Not that I am any official source of information, but here's my ten cents to this:
1) I would definitely ask the traineeship office!
2) As long as you haven't absolved one at an EU institution, you can always apply again.
3) I do not know about the exact rules, nor about which program type is the "easiest" to get into. But IMHO I don't think that there can be any rule prohibiting you from applying to another program, there only might be a rule saying that you can only apply to one program type at a time.
Hope this helped at least a little bit. Good luck!