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.
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.
If what you mean by "leaving your king open" is not moving it out of check if your opponent puts your king in check, then yes, that is an illegal move. If your king is in check you mustmove to protect it. If you mean simply that you leave your king in the center of the board without castling or other protection, then it is not an illegal move. It may (and probably will) result in your king coming under attack very quickly, however.
yes you do
by asking nicely! Check the instruction manual
You don't take/kill a king in chess. You have to put it into checkmate, ie, there are no moves the opponent can make with any piece, to take the king out of check. When this happens, the game is over.
You cannot capture opposite king in chess. You can attack him with check, and your opponent should immediately defend it. If the opponent has no way to defend, it is checkmate and you are awarded victory of the game.
Yes, as long as the move doesn't leave the King in Check from another source.
Yes, so long as the queen is not being protected by another piece or is more than one space away. The king cannot move into check, but if the queen is right in front of him, with nothing protecting, take her.
No, because before you kill the king, but you're about to after your opponent's turn, they must get out of check because your queen is checking the king. In other words, they must use their own chess pieces to block the queen or they can move the king into a different spot that is not checked.
Yes. A pawn may checkmate a king in the right situation. Note, however, that technically speaking a king is never "killed" but is checkmated. The game is over before the pawn actually captures or "kills" the king.