Place a RadComboBox in the GridViewColumn header and set SelectedIndex in the xaml. The SelectionChanged event is fired, but the command is not executed.

As a workaround you can set the SelectedIndex in xaml instead of on the Loaded event,

See the supported thread.

Create a RadComboBox, bound to a collection of objects, implementing INotifyPropertyChanged. Use DisplayMemberPath. Select an item, then change the display member property of the selected item - the selection box is not updated.
As workaround you can use DataTemplate:
<telerikInput:RadComboBox HorizontalAlignment="Left" Margin="122,52,0,0" VerticalAlignment="Top" MinWidth="150" x:Name="radComboBox" telerik:TextSearch.TextPath="Name">
    <telerikInput:RadComboBox.ItemTemplate>
        <DataTemplate>
            <TextBlock Text="{Binding Name}" />
        </DataTemplate>
    </telerikInput:RadComboBox.ItemTemplate>
</telerikInput:RadComboBox>