Mmm-kay, however noob this may sound, what do you do when:
A character with Override (say Lobot) moves adjacent to a closed door, which opens as his turn ends, and as the description of Override states "At the end of its turn, this character can designate 1 door that it can see as open or closed..." Can Lobot so designate a door within his line of sight at the end of his turn through the door that opened at the end of his turn?

End of turn effects are resolved in the order that the character's controller wishes. In fact, add it in like this. With Leia, Senator, he could move adjacent, do door effects, override another door open, and then move 2 squares. (Someone correct me if I'm getting this wrong.)

Thank you for your answer, sir: it makes sense to me, m'Lord Vader.

Yeah, you're correct. The trick is that you have to resolve ALL of Override at the same time. So, you have to both stop the old Override, and start the new Override, and THEN check your other doors, or vice versa. The hang up is where people try to insert other end of turn effects into the middle of the Override sequence, and that just doesn't work.