No, you do not have to say "check" before taking the king in chess.
No, you do not have to say "check" before declaring "checkmate" in a game of chess.
No, failing to say "check" when putting your opponent's king in danger does not result in an automatic loss in chess. It is not a rule in the game to verbally announce "check."
Before achieving checkmate in a game of chess, you need to check if your opponent's king is in a position where it cannot escape capture on the next move.
No, you cannot castle out of check in chess.
No, you are not allowed to castle out of check in chess.
No, you cannot castle when your king is in check in chess.
No, you cannot castle through a check in chess.
No, you cannot castle through check in chess.
No, it is not possible to castle out of check in a game of chess.
No, a king cannot check another king in chess.
No, you cannot castle in chess if your king is currently in check.