Correct. Kxf5 is not a legal move.
When a pinned piece is not pinned
My first post in the forum... This post is part sanity check that I understand the rules correctly, and part remarking that I didn't realize this could happen. See screenshot. I got this puzzle wrong because I was seeing the black rook as completely pinned and therefore unable to defend the f-pawn when it attacks the king. But I guess theoretically the white king cannot take the pawn because the black rook would capture the white king before vice versa. Thus for purposes of defending a checking or checkmating piece, a pinned piece is ... not pinned.
It is precisely because of this reason that the White king cannot capture the pawn. That king will get captured first before the Black king gets captured.
The Black rook is still considered pinned, but it can still prevent the opposing king from entering its path in its immobile state.
Soldiers who stand still in a position can still see what is happening around them even though they are standing still.
This "the King gets captured" talk is just enforcing this misunderstanding. The King never gets captured, it just gets attacked and even a pinned piece can do that.
From the official FIDE rules: "A piece is considered to attack a square, even if such a piece is constrained from moving to that square because it would then leave or place the king of its own colour under attack."

Checkmate and pins work where whoever can take the king wins the game, regardless of check. If White were to take f5, Black could capture White's king first, winning. However, any other piece could take f5 for free unless it's also pinned
I think the point for me here is keeping my mind flexible about what the pinned piece can still do, such that I don't miss a forcing move. It goes along with the tactic of unpinning the king to turn the pinned piece into an attacker (not possible in this puzzle, but something I learned about the hard way in a recent game).

My first post in the forum... This post is part sanity check that I understand the rules correctly, and part remarking that I didn't realize this could happen. See screenshot. I got this puzzle wrong because I was seeing the black rook as completely pinned and therefore unable to defend the f-pawn when it attacks the king. But I guess theoretically the white king cannot take the pawn because the black rook would capture the white king before vice versa. Thus for purposes of defending a checking or checkmating piece, a pinned piece is ... not pinned.
You got it, eventually. That's why you really need to closely at the board, as often the position isn't as "straightforward" as it may seem. Next time you'll remember this; it's a good learning point.

with great moustache, comes great responsibility
Field Marshal Seymon Budyonny was described by his own Chief-of-Staff as "a man with a huge mustache but a very small brain".

with great moustache, comes great responsibility
Field Marshal Seymon Budyonny was described by his own Chief-of-Staff as "a man with a huge mustache but a very small brain".
XD
The piece is pinned, it can not move. However, it still controls other squares while it's pinned.
I disagree. The pinned rook doesn't exert any meaningful control over the f file for any white piece other than the king. For me seeing the difference is part of the point here.
(The FIDE rules use 'attack' in an official way, but not 'control'.)
The easiest way to think about it is this - Which king gets captured first? Thus, a pinned piece can still give check, even if it is otherwise pinned.
In general, pinned pieces don't defend, except if it's the king trying to move into an attacked square. An attacked queen or bishop, in the same position as shown above, could capture the f5 pawn without fear.
By the way, in the puzzle, the puzzle solution appears to be 1. ..f5+ 2. Kg5 Qh6+ 3. Qxh6 gxh6+ 4. Kxh6 Rxd8 winning a rook. Is this correct?
... By the way, in the puzzle, the puzzle solution appears to be 1. ..f5+ 2. Kg5 Qh6+ 3. Qxh6 gxh6+ 4. Kxh6 Rxd8 winning a rook. Is this correct?
Yes. Not merely winning a rook, but getting out of mate-in-1 and ending up up-the-exchange for the endgame.

"Pinned" describes the reduction of a piece's ability to move along its eligible squares, not the removal of that eligibility.
The piece is pinned, it can not move. However, it still controls other squares while it's pinned.
I disagree. The pinned rook doesn't exert any meaningful control over the f file for any white piece other than the king. For me seeing the difference is part of the point here.
(The FIDE rules use 'attack' in an official way, but not 'control'.)
The words attack and control are used interchangeably in chess literature. Of course, feel free to create your own language.
If you need help, please contact our Help and Support team.
My first post in the forum... This post is part sanity check that I understand the rules correctly, and part remarking that I didn't realize this could happen. See screenshot. I got this puzzle wrong because I was seeing the black rook as completely pinned and therefore unable to defend the f-pawn when it attacks the king. But I guess theoretically the white king cannot take the pawn because the black rook would capture the white king before vice versa. Thus for purposes of defending a checking or checkmating piece, a pinned piece is ... not pinned.