When a player has moved some of their characters off of the board, do the treasures on those characters still count towards missions? (I assumed that they do.)

Also, when a player has moved ALL of their characters off of the board, do they still get a turn as usual? If so, can they still perform missions? Do they move a henchman as usual? Can they play Arcana cards that affect other players as usual?

Thanks.

Good question. And, theoretically, is there anything stopping a character who has left the board from coming back into the neighbourhood to ****** one last gem?