Yes, the king can capture its attacker as long as the the attacking piece is adjacent to the king and if the king can move to the space where the attacker is without putting himself in check from some other piece of the opponent.
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Technically, mate is the capture of the king, but the piece cannot be taken and removed from the board in the same way as any other piece can.
so tecnically you can never kill eat nor take the king of the boardNo. the king may only move one space at a time (aside from castling). This being the case, a king may not move into a check position, so could not be placed in a position to be taken by another king.