Can a pinned piece deliver checkmate?

Yes. Nothing about a pin changes the squares a piece attacks, so a pinned rook can give the final check and a pinned bishop can cover the last flight square. A piece with no legal moves at all can still be the reason a mate works, which is a fair thing to check for before you resign.

Can a pinned piece capture the piece that is pinning it?

Usually yes, and it is often the answer. Capturing the pinning piece is a move along the line of the pin, so it does not expose your king. A rook pinned down the d-file may take the rook on d8; a knight pinned to the king cannot, because a knight can never travel along a line.

Does a pinned pawn still control the squares it attacks?

Yes. A pawn on f2 pinned down the f-file may not capture on e3 or g3, because that would leave its own king in check, but it still attacks both squares. An enemy king may not walk onto them, and an enemy piece that lands there can be taken the moment the pin is broken.

Can a pinned piece give check?

The white knight on e4 is checking the black king on f6, and that knight cannot move a muscle. The rook on e8 is staring straight down the e-file at the white king on e1, so every square the knight could jump to would leave its own king in check. Pinned, frozen, no legal moves whatsoever. Black is still in check and still has to do something about it.

Black to move and in check from a knight that has nowhere to go. The rook on e8 pins it to the king on e1. Black answers the check the cheapest way, 1...Rxe4+, and the pinned knight never got to move.

Moving and attacking are two different things

A pin restricts where a piece may go. It says nothing at all about what the piece hits, and the Laws are deliberate about this: a piece counts as attacking a square even when it could not legally move there, because moving would expose its own king. So a pinned knight still covers every square it attacks, a pinned bishop still owns its diagonal, and a pinned pawn still guards the two squares diagonally in front of it.

That last one catches everybody. A white pawn on f2 pinned down the f-file may not capture on e3 or g3, but it still stops the black king setting foot on either.

Which is exactly how it costs you a game

The thought goes like this: his knight is pinned, so it does not count, so my king can go there. It counts. A king may not move onto a square attacked by any enemy piece, pinned or otherwise, and it may not capture a piece that a pinned piece defends. Calculate a king walk with a pinned defender written off and you are calculating a different position from the one on the board. There is more on that trap on whether a king can move into check.

It runs the other way too. Your own pinned piece is still doing defensive work, so do not swap it off on the grounds that it cannot move anyway. It is holding squares while it stands there.

A pinned piece can be the one that mates

Count the bishop's legal moves before reading on. It has none: the rook on b7 pins it to the king on b1 and every square a bishop on b2 can reach is off the b-file. It still covers g7, the one flight square the rook cannot take, so 1.Rd8 is mate.

The rook checks along the eighth rank and covers f8 and h8 itself. The black king’s only other door is g7, and it is shut by a bishop that is nailed to the board. A piece with zero legal moves decided the game.

One more thing a pinned piece can do: move along the line of the pin, and give check from the new square. White king on b1, white queen on d3, a black bishop on f5 pinning her along f5-e4-d3-c2-b1. She is pinned, but Qe4 stays on that diagonal, so it is legal, and if the black king is sitting on e8 then Qe4 is check. The pin never entered into it.

Where the pin does bite is the moment you want the piece to leave the line, and that is a separate question with a two-part answer: can you move a pinned piece. If you would rather stop walking into pins than read about them, pin puzzles are the drill.