Instead of my typing I find seeing is easier.

Yes, as you say: other pieces can take a pawn straight on.
The pawn is the only chessman that does not capture the same way that it moves.
Your example is correct: where you described a pawn is right next to a rook,
the pawn cannot take the rook, but the rook take the pawn.

yes the rook could take a pawn sitting directly next to it as long as it is to the right left front or back. queen could take a pawn no matter which square diagonal or straight or side as could the king. the bishop only takes on the diagonal and the knight in an L shape. If the other player puts himself in check – then you can take him and that is just stupid – but good manners would require that you say check – if he has not removed his fingers from the piece yet – he can change the move

well, for your first question, the rook can take the pawn. why? because a rook can move horizontically ang diagonally and even somewhat “eat” the opponent’s piece.
For your second question no. It’s because you will give the opponent a hint that you are going to place a check on his king. You will only say “check” when you have already checked the opponent’s king.
For your third question, you cannot place yourself in check. That’s wrong. But then if you play a real live game of chess, that could be possible. you don’t have to do anything. just tell your opponent that that move made your king checked.

no the rook kannot . unless u feel guilty u gotta say sumthing but just keep quite theres no rules 4 putting urself in check soo just kill the king!!