Neither, he probably used a special move to capture your pawn "in passing." See En Passant for more details.
You may have stalemated your opponent. See Stalemate for more details,
You likely stalemated your opponent instead of checkmating him. See Checkmate vs.Stalemate for more details.
He promoted a pawn to a new queen. See Promotion for more details.
There are several things that can prevent you from castling. You may have moved your king before, or you may be in check or trying to move through check. See Castling for a complete description.
Yes, you are permitted to castle when your rook is under attack. See Castling for full details.
A pinned piece can still deliver check or checkmate. It doesn't matter if it can "move" or not. See Checkmate Misconceptions for more details.
You (and your opponent) have enabled the "king capture" option. This option emulates over-the-board games by allowing you to move into check and allowing your opponent to capture your king to claim a win. You can identify when this option is active by the small "KC" icon near the move list.