Yes, the king can take another piece to get out of check, but only to get out of check.
Yes, as long as it doesn't put him in check.
The objective of chess is to checkmate the king, where the king is in check by a piece and it cannot block the check, move to another square, or capture the piece checking the king.
The King is not allowed to be in check. If a piece is on a square where if the King were there, the King would in check, the King isn't allowed to capture it.
No , your King must be out of check . You can block a check with another piece - that piece in turn can place your opponent's King in check ,
Your next move must take the king out of check, either by moving him out of threat, by blocking it with another piece, or by taking the attacking piece. A king cannot castle out of check.
The king can take another piece anytime, but it is not smart to do so until perhaps the last moves of the game, if at all.
Yes. The only way is to move a piece so that it blocks your king from check and checks or attacks another piece of the opponent. Another way is for the king itself to attack the piece that is attacking it. Example is if the queen foolishly moves adjacent to the king without being protected by another piece, the king may capture the queen. It is not limited to simply moving out of check.
Only in certain circumstances such as having your king in check with no squares to move to and the only way to stop the check is to capture the piece delivering the check.
The king can never be 'taken' by an opposing piece. When the King is under threat from another piece, the next move must be to get it out of check, either by moving the King, or breaking check by interposing another piece in between King and aggressor, or by taking the opposing piece. If the check cannot be broken, then it is Checkmate, and the game ends. A common gesture of surrender though, is to knock one's king over to decare you are resigning from the game.
No. Most people get weaker things farther if another piece is protecting it, and it is adjacent to the king.