To reproduce the case with the double click:

AddDropDownList();
radDropDownList1.SelectNextOnDoubleClick = true; 
radDropDownList1.ReadOnly = true;

Start the app and double click the test area

To reproduce the case with the first letter: 

AddDropDownList();
radDropDownList1.ReadOnly = true;
radDropDownList1.DropDownStyle = RadDropDownStyle.DropDownList;

Start the app and press the first letter of some of the items in the control

Workaround: http://www.telerik.com/community/forums/raddropdownlist-readonly-not-working-with-selectnextondoubleclick