May it in a puzzle? (White to move and WIN)

Not sure: is it a valid puzzle? (in all aspects)

White to move and WIN

 ( being the solution:       1. gxh6 (e.p.)   Bc2            2. h7  Bxh7        3. O-O-O#    )

1. It cannot be proven that h7-h5 was the last move, so by convention gxh6 e.p. is illegal. You have to make sure h7-h5 was the last move in order to make en passant legal. Note that the reverse convention applies for castling: if you cannot prove that castling is illegal, then castling is allowed by convention.

2. It is not a mate in 3 for obvious reasons, but since the stipulation is White to win, okay... What happens after 1... Rxg3 2. hxg3 h2 and so on, where Black tries to promote h-pawn too?

Every problemist's got to know a little bit of retro. At least for legality/illegality, castling, and en passant. The Codex, article 16.

As for soundness... going with the anti-convention for a bit, 1. gxh6 e.p.!? Rxg3 2. hxg3 h2 3. Kd2+ Kg2 4. h7 h1=Q 5. Rxh1 etc is winning. Instead interesting is 1...Kh1 2. h7 (2. Ra3 Bc2) Rg1+ 3. Kf2 Rxa1 and black may have some kind of fortress going on. Or not; probably still a clear white win due to h3 dropping eventually.

1. thx, but how to make sure h7-h5 was the last move?

2. after 1... Rxg3 2. hxg3 h2   follows: Ke2+ (or Kd2+) , and the rook attacks h1

To Remellion, in both lines you presented, white wins easilly. In the 1st, you showed it, and in the second, after 1. gxh6 e.p., we have the following :   1... Kh1   2. h7 Rg1+   3. Kf2 Rxa1   4. h8=Q  , and White threatens Qa8.   

This is another variant where White, fearing 2. ... Rg1, makes a mistake, only in such hyphotesis Black  survives or wins:

Last post, what???

2. a7 wins Rxg3 is not a problem and losing the rook and getting a queen is good

1. You cannot do it easily. That's when you start entering the field of retrograde analysis. To make sure a double-step pawn was the last move, you need to arrange things carefully. Here's one example just to show how to force the last move, but this is not intended to be a directmate or anything.

If it's White to move, then Black made the last move, and it must be h7-h5.

Ach, missed Qa8 - funny how long queen moves are easy to overlook.

To force en passant is usually tricky, requiring that most of the composition is dedicated to it. I have a thread "Do you know checkmate?" where the first position is a forced en passant mate; it's pretty tricky to make these.
The record for smallest composition goes to F. Amelung in 1897 (!). Mate in 2.