Yes, as long as the move doesn't leave the King in Check from another source.
Yes, the king can take another piece to get out of check, but only to get out of check.
The king can move one square in any direction. If it moves into a square that is occupied by another piece, it takes that piece only if the king is not endangered if it gets next to it in order to capture it.
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, it can kill, or rather capture/checkmate a king in chess. A pawn can capture any other piece on the board. A king can be checkmated by any other piece on the board except by the other 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.
You can't checkmate another king unless the king is the only piece left on the board.
The king just moves onto it, same as any other capture. The difference is that the king cannot capture a piece or pawn protected by another piece or pawn, as this would place him in check (attacked, subject to capture).
The king (if possible) can move one space in any direction. If a piece happened to be in king's movement area, it can be killed by the king.
Yes, in fact if you're trapped you can move the top piece of the king with another piece.