No. In addition to moving the King, check can be escaped by capturing the checking piece or blocking its path. Of course a Knight cannot be blocked, so that option is not available in that case. Also note that the King cannot avoid check by "castling".
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The king can move at any time as long as the square it goes to does not put it in check. If the king were to be put into check, it has three options:
1) Move from the square to a square where it isn't put in check.
2) Block the check with another piece (unless the attacking piece is a knight).
3) Capture the attacking piece.
No. Most people get weaker things farther if another piece is protecting it, and it is adjacent to the king.
If the capture takes the king out of check and it does not become in check from a different source, then yes.