Implement the option that will display "No Results Found" in case no items are returned after the request.

The "Show All Results" is Searchbox cannot be accessed with a keyboard when navigating through the items in the dropdown. 

Validation is not implemented for RadSearchBox.

You can find attached an example that emulates validation of a search box through a dummy textbox and validators attached to that dummy textbox.

When the SearchBox is on the right side of the screen and the dropdown width is larger than the SearchBox width, the dropdown goes outside the viewport:
Actual: https://www.screencast.com/t/4SoNcRCj
Expected/fixed: https://www.screencast.com/t/smfPgy767M

Workaround: 

<script>
    function OnClientLoad(sender, args) {
        sender._dropDown.on({ "reflowed": sender._onReflowed }, sender)
    }

    Telerik.Web.UI.RadSearchBox.prototype._onReflowed = function () {
        var that = this;
                       
        var $animationContainer = $(that._dropDown._animationContainer);

        if (that._enableScreenBoundaryDetection) {
            var availableSpace = that._dropDown._getAvailableSpace();
            var tableWidth = $(that._innerWrapElement).outerWidth();
            var dropDownWidth = $animationContainer.outerWidth();
            var overflow = dropDownWidth - (tableWidth + availableSpace.right);

            if (overflow > 0) {
                var left = parseInt($animationContainer.css("left"), 10);
                var newLeft = Math.max(left - overflow, left - (dropDownWidth - tableWidth));

                $animationContainer.css("left", newLeft);
            }
        }
    }
</script>
<telerik:RadSearchBox runat="server" ID="RadSearchBox1"
    DataTextField="ShipName" OnClientLoad="OnClientLoad" DropDownSettings-Width="500px" DataValueField="OrderID">        
</telerik:RadSearchBox>

Create an event that will be triggered after items have been requested.