No. You may not move INTO check. If that is the only move available, it would then depend whether or not you are already in check.
If you are in check and cannot move to a square where you are not in check, it is checkmate, the game is over, and you have lost.
If you are not in check, but cannot move without going into check, and you have
no other piece or pawn that can move, then it is stalemate, and the game is a draw.
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It is illegal to move your king into checkmate because that's like the same as moving your king into check, which is not illegal. If they check you, you must go out of check as in game rules. If you go out of check and into check, you must choose a different path and redo that move.
Yes. The move would have to protect your king and put the other player into checkmate though.