In Attempt 1 you state "but then focus is lost so the autocomplete box doesn't display". This sounds like maybe the click happened at the wrong coordinates. Can you verify the mouse is moving to the right coordinates before it clicks? If <RadListBox>.Items gives us incorrect coordinates this problem can happen.
Attempt 2 sounds like it may be the same problem as attempt 1.
In attempt 3, is SimulateRealTyping checked for the Type "Test" into the box step? This is very important for autocomplete controls.