Use attached to reproduce!

Workaround 
The column name must match the field name.

To reproduce: 
- Set the DropDownStyle to DropDownList

To reproduce: run the attached project and refer to the attached gif file.

Workaround:

 this.radMultiColumnComboBox1.MultiColumnComboBoxElement.TextBoxElement.TextBoxItem.TextBoxControl.MouseDown += TextBoxControl_MouseDown;

        private void TextBoxControl_MouseDown(object sender, MouseEventArgs e)
        {
            this.radMultiColumnComboBox1.MultiColumnComboBoxElement.TextBoxElement.TextBoxItem.TextBoxControl.SelectAll();
        }

Workaround: 
1. Position the RadMultiColumnComboBox control and instead of setting the Dock property to Fill, anchor the it on all four sides.
Alternatively:
2. Set the Dock property to Fill, but also set a MinimumSize of the column defining a height

To reproduce:

            this.radMultiColumnComboBox1.DataSource = this.customersBindingSource;
            this.radMultiColumnComboBox1.DisplayMember = "CustomerID";
            this.radMultiColumnComboBox1.AutoFilter = true;

            FilterDescriptor filter = new FilterDescriptor();
            filter.PropertyName = this.radMultiColumnComboBox1.DisplayMember;
            filter.Operator = FilterOperator.Contains;
            this.radMultiColumnComboBox1.EditorControl.MasterTemplate.FilterDescriptors.Add(filter);

1. Click the dropdown button, verify that all items list on drop down
2. Type into the textbox so only one item is left( e.g. "BERGS" )
3. While the drop-down is still opened, click the arrow button.
4. Focus another control on the form and then when the user clicks the arrow button, only one item is still shown in the drop down. 

Workaround:
        private void radMultiColumnComboBox1_DropDownClosed(object sender, RadPopupClosedEventArgs args)
        {
            this.radMultiColumnComboBox1.EditorControl.MasterTemplate.Refresh();
        }

To reproduce: please refer to the attached sample project which result is illustrated in the attached gif file. The DropDownClosed is expected to be fired when the drop down is already closed. Hence, showing a message box shouldn't keep the drop down opened. The problem is reproducible with RadDropDownList as well. Note that this problem is not applicable for MS ComboBox.

Workaround:  this.radMultiColumnComboBox1.MultiColumnComboBoxElement.DropDownAnimationEnabled = false;

1. Create a project with RadGridView
2. Add a GridViewMultiComboBoxColumn
3. Attach to CellEditorInitialized event and try to handle PopupOpened event to change the popup size.

To reproduce: please refer to the attached sample project and refer to the screenshot. You will notice that initially, the arrow is a few pixels up. If you disable and then enable the RadMultiColumnComboBox again you will notice the difference.
Workaround:

        private void radToggleButton1_ToggleStateChanged(object sender, Telerik.WinControls.UI.StateChangedEventArgs args)
        {
            ImagePrimitive img = this.radMultiColumnComboBox1.MultiColumnComboBoxElement.ArrowButton.FindDescendant<ImagePrimitive>();
            img.Margin = new Padding(0);
            if (args.ToggleState == Telerik.WinControls.Enumerations.ToggleState.On)
            {
                this.radMultiColumnComboBox1.Enabled = true;
                this.radMultiColumnComboBox1.MultiColumnComboBoxElement.ArrowButton.Padding = new Padding(0);
            }
            else
            {
                this.radMultiColumnComboBox1.Enabled = false; 
            }
        }

To reproduce: please refer to the attached sample project and try to enter some numeric value or clear the existing one. Then open the popup.

Workaround: use the custom filtering that RadGridView offers http://docs.telerik.com/devtools/winforms/gridview/filtering/custom-filtering

this.radMultiColumnComboBox1.AutoFilter = true;
FilterDescriptor filter = new FilterDescriptor();
filter.PropertyName = this.radMultiColumnComboBox1.DisplayMember;
filter.Operator = FilterOperator.Contains;
this.radMultiColumnComboBox1.EditorControl.MasterTemplate.FilterDescriptors.Add(filter);
this.radMultiColumnComboBox1.EditorControl.EnableCustomFiltering = true;
this.radMultiColumnComboBox1.EditorControl.CustomFiltering += EditorControl_CustomFiltering;

string search = string.Empty;

private void EditorControl_CustomFiltering(object sender, Telerik.WinControls.UI.GridViewCustomFilteringEventArgs e)
{
    search = this.radMultiColumnComboBox1.MultiColumnComboBoxElement.EditorElement.Text;
    if (search != string.Empty)
    {
        e.Handled = true;
        e.Visible = e.Row.Cells[this.radMultiColumnComboBox1.DisplayMember].Value.ToString().Contains(search);
    }
    else
    {
        e.Handled = false;
    }
}

To reproduce: 

            DataTable dt = new DataTable();

            dt.Columns.Add("ID");
            dt.Columns.Add("Description");

            dt.Rows.Add(new object[] { "0", "Low" });
            dt.Rows.Add(new object[] { "1", "Medium" });
            dt.Rows.Add(new object[] { "2", "High" });
            radMultiColumnComboBox1.DisplayMember = "ID";
            radMultiColumnComboBox1.ValueMember = "ID";
            radMultiColumnComboBox1.DataSource = dt;
            radMultiColumnComboBox1.SelectedValue = "1";

 this.radMultiColumnComboBox1.SelectedValueChanged+=radMultiColumnComboBox1_SelectedValueChanged;

        private void radMultiColumnComboBox1_SelectedValueChanged(object sender, EventArgs e)
        {
            Console.WriteLine(this.radMultiColumnComboBox1.SelectedValue);
        }

If you type 2 in the editable part, the SelectedValue is not changed as in the previous version.

Workaround:

public class MyRadMultiColumnComboBox : RadMultiColumnComboBox
{
    public override string ThemeClassName
    {
        get
        {
            return typeof(RadMultiColumnComboBox).FullName;
        }
    }

    protected override RadMultiColumnComboBoxElement CreateMultiColumnComboBoxElement()
    {
        return new MyRadMultiColumnComboBoxElement();
    }
}

public class MyRadMultiColumnComboBoxElement : RadMultiColumnComboBoxElement
{
    protected override Type ThemeEffectiveType
    {
        get
        {
            return typeof(RadMultiColumnComboBoxElement);
        }
    }

    protected override void SetActiveItem(string text)
    {
        int rowIndex = this.FindItemIndexExact(text);

        if (rowIndex != -1)
        {
            this.EditorControl.CurrentRow = this.EditorControl.Rows[rowIndex];
            this.textBox.SelectionStart = this.textBox.Text.Length;
        }
    }
}

To reproduce, add RadMultiColumnCombobox on a form and use the following code:

  public Form1()
        {
            InitializeComponent();

            Random r = new Random();
            DataTable table = new DataTable();

            table.Columns.Add("ID", typeof(int));
            table.Columns.Add("Name", typeof(string));
            table.Columns.Add("Bool", typeof(bool));
            table.Columns.Add("DateColumn", typeof(DateTime));

            for (int i = 0; i < 10; i++)
            {
                table.Rows.Add(i, "Row " + i, r.Next(10) > 5 ? true : false, DateTime.Now.AddHours(i));
            }

            radMultiColumnComboBox1.DataSource = table;
            radMultiColumnComboBox1.DisplayMember = "Name";
            radMultiColumnComboBox1.AutoFilter = true;
            radMultiColumnComboBox1.DisplayMember = "Name";
            FilterDescriptor filter = new FilterDescriptor();
            filter.PropertyName = radMultiColumnComboBox1.DisplayMember;
            filter.Operator = FilterOperator.Contains;
            radMultiColumnComboBox1.EditorControl.MasterTemplate.FilterDescriptors.Add(filter);
        }

Once the form loads up, select the text in the editable area and enter any number from 0 to 9. It is important the width of the MCCB is smaller than the width the columns take.

FIX. RadMultiColumnComboBox - AutoFilter does not work in unbound mode

To reproduce drop RadMultiColumnComboBox on a form and use the code below:

 public Form1()
        {
            InitializeComponent();

            ThemeResolutionService.AllowAnimations = false;
    
            RadMultiColumnComboBoxElement multiColumnComboElement = this.combo1.MultiColumnComboBoxElement;
            multiColumnComboElement.DropDownSizingMode = SizingMode.UpDownAndRightBottom;
            multiColumnComboElement.EditorControl.MasterTemplate.AutoGenerateColumns = false;
            multiColumnComboElement.EditorControl.AutoGenerateColumns = false;

            GridViewTextBoxColumn column1 = new GridViewTextBoxColumn("Item");
            column1.HeaderText = "Item";
            multiColumnComboElement.Columns.Add(column1);

            GridViewTextBoxColumn column2 = new GridViewTextBoxColumn("Description");
            column2.HeaderText = "Description";
            multiColumnComboElement.Columns.Add(column2);

            combo1.DisplayMember = "Description";
            combo1.ValueMember = "Item";

            this.combo1.AutoSizeDropDownToBestFit = true;
            this.combo1.DataSource = GetData();

            FilterDescriptor filter = new FilterDescriptor();
            filter.PropertyName = this.combo1.DisplayMember;
            filter.Operator = FilterOperator.StartsWith;
            this.combo1.EditorControl.MasterTemplate.FilterDescriptors.Add(filter);
            this.combo1.AutoFilter = true;
            this.combo1.MultiColumnComboBoxElement.AutoCompleteMode = AutoCompleteMode.Suggest;
            this.combo1.DropDownStyle = RadDropDownStyle.DropDown;
            this.combo1.SelectedItem = null;
        }

        private List<CustomItem> GetData()
        {
            List<CustomItem> items = new List<CustomItem>();

            for (int i = 0; i < 20; i++)
            {
                CustomItem item = new CustomItem(i, String.Format("This is item number: {0}", i.ToString()));
                items.Add(item);
            }

            return items;
        }

    }

    [Serializable]
    public class CustomItem
    {
        public int Item { get; set; }
        public string Description { get; set; }

        public CustomItem(int pItem, string pDescription)
        {
            this.Item = pItem;
            this.Description = pDescription;
        }
    }

Workaround: 
1. Enable the animations
 ThemeResolutionService.AllowAnimations = true;
or 
2. Increase the popup size by one pixel when opening:
 void multiColumnComboElement_PopupOpening(object sender, CancelEventArgs e)
        {
            RadMultiColumnComboBoxElement multiColumnComboElement = (RadMultiColumnComboBoxElement)sender;
            multiColumnComboElement.MultiColumnPopupForm.Size = new Size(multiColumnComboElement.MultiColumnPopupForm.Size.Width + 1, multiColumnComboElement.MultiColumnPopupForm.Size.Height);
        }
or 
3. Turn off the control clipping
((RadHostItem)((MultiColumnComboPopupForm)multiColumnComboElement.MultiColumnPopupForm).SizingGripDockLayout.Children[1]).ClipControl = false;

How to reproduce: check the attached project and video

Workaround: create a custom RadMultiColumnComboBoxElement
Public Class MyRadMultiColumnComboBox
    Inherits RadMultiColumnComboBox

    Public Overrides Property ThemeClassName As String
        Get
            Return GetType(RadMultiColumnComboBox).FullName
        End Get
        Set(value As String)
            MyBase.ThemeClassName = value
        End Set
    End Property

    Protected Overrides Function CreateMultiColumnComboBoxElement() As RadMultiColumnComboBoxElement
        Return New MyRadMultiColumnComboBoxEleemnt()
    End Function
End Class

Class MyRadMultiColumnComboBoxEleemnt
    Inherits RadMultiColumnComboBoxElement

    Protected Overrides ReadOnly Property ThemeEffectiveType() As Type
        Get
            Return GetType(RadMultiColumnComboBoxElement)
        End Get
    End Property

    Protected Overrides Sub SetActiveItem(item As String)
        Dim rowIndex As Integer = Me.FindItemIndexExact(Text)

        If rowIndex <> -1 Then
            Me.textBox.SelectionStart = Me.textBox.Text.Length
        End If
    End Sub
End Class

Workaround: create a custom RadMultiColumnComboBox control with a custom element. You can also check the attached project.

public class MyRadMultiColumnComboBox : RadMultiColumnComboBox
{
    public override string ThemeClassName
    {
        get
        {
            return typeof(RadMultiColumnComboBox).FullName;
        }
    }

    protected override RadMultiColumnComboBoxElement CreateMultiColumnComboBoxElement()
    {
        return new MyRadMultiColumnComboBoxElement();
    }
}

public class MyRadMultiColumnComboBoxElement : RadMultiColumnComboBoxElement
{
    protected override Type ThemeEffectiveType
    {
        get
        {
            return typeof(RadMultiColumnComboBoxElement);
        }
    }

    protected override void ProcessKeyDown(object sender, KeyEventArgs e)
    {
        base.ProcessKeyDown(sender, e);

        if (this.IsPopupOpen && (e.KeyCode == Keys.PageDown || e.KeyCode == Keys.PageUp))
        {
            this.EditorControl.GridBehavior.ProcessKey(e);
        }
    }
}

When you set the DropDownHeight property to a big number that exceeds the screen's height, the applied values is SystemInformation.WorkingArea.Height - 100 . However, it should be only SystemInformation.WorkingArea.Height. These 100 pixels shouldn't be lost.

NOTE: the same behavior is observed with DropDownWidth as well.

Workaround:

Public Class CustomMultiColumn
Inherits RadMultiColumnComboBox

    Protected Overrides Function CreateMultiColumnComboBoxElement() As RadMultiColumnComboBoxElement
        Return New CustomRadMultiColumnComboBoxElement()
    End Function

    Public Overrides Property ThemeClassName As String
        Get
            Return GetType(RadMultiColumnComboBox).FullName
        End Get
        Set(value As String)
            MyBase.ThemeClassName = value
        End Set
    End Property
End Class

Public Class CustomRadMultiColumnComboBoxElement
Inherits RadMultiColumnComboBoxElement
    Protected Overrides Function GetPopupSize(popup As RadPopupControlBase, measure As Boolean) As Drawing.Size
        Dim s As Size = MyBase.GetPopupSize(popup, measure)
        Return New Size(s.Width, Math.Min(SystemInformation.WorkingArea.Height, Me.DropDownHeight))
    End Function

    Protected Overrides ReadOnly Property ThemeEffectiveType As Type
        Get
            Return GetType(RadMultiColumnComboBoxElement)
        End Get
    End Property
End Class

To reproduce:
Create a create a class with Properties Id, Name, ParentId

this.multiColumnComboBox.MultiColumnComboBoxElement
    .EditorControl.Relations.AddSelfReference(this.multiColumnComboBox.MultiColumnComboBoxElement.EditorControl.MasterTemplate, "Id", "ParentId");

this.multiColumnComboBox.DataSource = this.people;
this.multiColumnComboBox.DisplayMember = "Name";
this.multiColumnComboBox.ValueMember = "Id";

this.multiColumnComboBox.AutoFilter = true;
this.multiColumnComboBox.DisplayMember = "Name";
FilterDescriptor filter = new FilterDescriptor();
filter.PropertyName = this.multiColumnComboBox.DisplayMember;
filter.Operator = FilterOperator.Contains;
this.multiColumnComboBox.EditorControl.MasterTemplate.FilterDescriptors.Add(filter);

When you open the popup exception should occur.

To reproduce:
Create a ListView or GridView cell element. For example:
public class FilterGroupByCell : DetailListViewDataCellElement
    {
        private RadMultiColumnComboBoxElement multiColumnComboBox = new RadMultiColumnComboBoxElement();
        public FilterGroupByCell(DetailListViewVisualItem owner, ListViewDetailColumn column) :
            base(owner, column) { }

        protected override void CreateChildElements()
        {
            base.CreateChildElements();

            this.multiColumnComboBox.AutoSizeDropDownToBestFit = true;
            multiColumnComboBox.DropDownSizingMode = SizingMode.UpDownAndRightBottom;
            multiColumnComboBox.EditorControl.MasterTemplate.AutoGenerateColumns = false;

            GridViewTextBoxColumn column = new GridViewTextBoxColumn("CustomerID");
            column.HeaderText = "Customer ID";
            multiColumnComboBox.Columns.Add(column);
            column = new GridViewTextBoxColumn("ContactName");
            column.HeaderText = "Contact Name";
            multiColumnComboBox.Columns.Add(column);
            column = new GridViewTextBoxColumn("ContactTitle");
            column.HeaderText = "Contact Title";
            multiColumnComboBox.Columns.Add(column);
            column = new GridViewTextBoxColumn("Country");
            column.HeaderText = "Country";
            multiColumnComboBox.Columns.Add(column);
            column = new GridViewTextBoxColumn("Phone");
            column.HeaderText = "Phone";
            multiColumnComboBox.Columns.Add(column);

            this.Children.Add(multiColumnComboBox);
        }

        public override void Synchronize()
        {
            base.Synchronize();
            this.Text = "";

            if (base.Row == null)
            {
                return;
            }

            this.CurrentFilter = (MFilter)base.Row.DataBoundItem;

            if (CurrentFilter.CustomListRecordId == "1")
                multiColumnComboBox.Visibility = Telerik.WinControls.ElementVisibility.Collapsed;

            DataTable table1 = new DataTable("test");
            table1.Columns.Add("CustomerID");
            table1.Columns.Add("ContactName");
            table1.Columns.Add("ContactTitle");
            table1.Columns.Add("Country");
            table1.Columns.Add("Phone");

            table1.Rows.Add(1, "Ivan Petrov", "2", "2", "2");
            table1.Rows.Add(2, "Stefan Muler", "2", "2", "2");
            table1.Rows.Add(3, "Alexandro Ricco", "2", "2", "2");

            multiColumnComboBox.DataSource = table1;
            multiColumnComboBox.DisplayMember = "ContactName";
        }

        public MFilter CurrentFilter { get; set; }
    }

Workaround:
Create a custom RadMultiColumnComboBoxElement:
public class MyRadMultiColumnComboBoxElement : RadMultiColumnComboBoxElement
{
    protected override void OnParentChanged(Telerik.WinControls.RadElement previousParent)
    {
        if (this.IsPopupOpen)
        {
            this.ClosePopup(RadPopupCloseReason.CloseCalled);
        }

        if (this.Parent != null && this.Parent.ElementTree != null && this.Parent.ElementTree.Control is RadGridView)
        {
            base.OnParentChanged(previousParent);
        }
    }

    protected override Type ThemeEffectiveType
    {
        get
        {
            return typeof(RadMultiColumnComboBoxElement);
        }
    }
}

In order to be able to set a DataSource you need to create a new BindingContext prior setting setting the DataSource:

DataTable table1 = new DataTable("test");
table1.Columns.Add("CustomerID");
table1.Columns.Add("ContactName");
table1.Columns.Add("ContactTitle");
table1.Columns.Add("Country");
table1.Columns.Add("Phone");

table1.Rows.Add(1, "Ivan Petrov", "2", "2", "2");
table1.Rows.Add(2, "Stefan Muler", "2", "2", "2");
table1.Rows.Add(3, "Alexandro Ricco", "2", "2", "2");

multiColumnComboBox.EditorControl.BindingContext = new System.Windows.Forms.BindingContext();
multiColumnComboBox.DataSource = table1;
multiColumnComboBox.DisplayMember = "ContactName";

To reproduce:

public Form1()
{
    InitializeComponent();

    DataTable dt = new DataTable();
    dt.Columns.Add("Id", typeof(int));
    dt.Columns.Add("Name", typeof(string));
    for (int i = 0; i < 20; i++)
    {
        dt.Rows.Add(i, "Item" + i);
    }
    this.radMultiColumnComboBox1.DataSource = dt;
    this.radMultiColumnComboBox1.DisplayMember = "Name";
    this.radMultiColumnComboBox1.ValueMember = "Id";
    this.radMultiColumnComboBox1.SelectedValueChanged += radMultiColumnComboBox1_SelectedValueChanged; 
}

private void radMultiColumnComboBox1_SelectedValueChanged(object sender, EventArgs e)
{ 
    if (this.radMultiColumnComboBox1.SelectedIndex > -1)
    {
        MessageBox.Show("Test");
    }
}

NOTE: If you use a RadMessageBox, after showing the message the first time, the popup can't be closed because InvalidOperationException is thrown.

Workaround: show the message box in the RadMultiColumnComboBox.DropDownClosed event.

Steps to reproduce:
1. Select the radMultiColumnsComboBox1 at Design time
2. Press F4 for showing Properties
3. Open Columns editor (GridViewDataColumn Collection Editor)
4. Try to add a column
Error Message: Cannot create an instance of the abstract class or interface 'Telerik.WinControls.UI.GridViewDataColumn' because it is an abstract class.