yes
Yes, a pawn can put a king in check in a game of chess by moving diagonally to attack the king if it is in its path.
Yes, a pawn can check a king in chess if it reaches the opponent's back rank and promotes to a queen, rook, bishop, or knight.
A promoted pawn can move on it's next turn .
Yes, a pawn may capture any piece on the board as long as it is a legal move. One way it could happen is the pawn is blocking lets say a Bishop or a Rook from attacking the opponent's king. The pawn moves to a square where it attacks the Queen. Normally the Queen would either just capture the pawn or move away from it. But if the move of the pawn now places the king in check from that Bishop or Rook (this is called a discovered check) the King must move out of check. If the player now in check has no alternative but to move the King out of check, then once he moves the king, the pawn is free to capture the Queen.
No , only upon the pawn's promotion to a piece that then checks the king may you then say "check" .
The pawn can checkmate the king but only in conjunction with at least one other piece to protect the pawn from capture by the king and at least one enemy piece placed near the king so that it cuts off any escape avenues the king would have. If a pawn places a king in check and the king is completely surrounded by his own pieces in such a way that its only move would be capture the pawn and if no enemy piece is placed so that it can capture that pawn and if the pawn is protected from capture by one of its own pieces, the pawn has checkmated the king.
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.
For example if you want to drop a pawn to check the king but not to mate him (you cannot mate with pawn drops).
It really depends on what you're asking. When in check, the only legal moves are moves that would serve to get you out of check. If capturing the rook with your pawn would mean your king was no longer in check, then yes, the rook may be taken. If capturing the rook with your pawn would mean that your king would be in check by another piece, then no, the rook may not be taken.
No, a pawn in chess cannot take a king. The game ends when the king is in checkmate, not captured by a pawn.
Yes , and your opponent must counter the move .
No, a pawn cannot capture a king in chess.