2)"Can you think of any positions in which castling would be the only legal move (illegal in a no-castling variant)?"
It is pretty obvious that castling is never the only legal move: if you can castle you are not in check, and you can move the King one step towards the Rook, and the Rook anywhere between Rook and King. Unless you are talking about Chess960 and Fischer castling:
It is pretty obvious that castling is never the only legal move: if you can castle you are not in check, and you can move the King one step towards the Rook, and the Rook anywhere between Rook and King. Unless you are talking about Chess960 and Fischer castling: