You played a very quiet and unambitious opening, but you were still dreaming of a King-side attack. You might read Vukovic The Art of Attack in Chess and consider his assertion that a successful attack depends on certain "preconditions" being met. You had none of those preconditions, and you should have been focussed merely on maintaining a sound, active position without weaknesses.
I do not understand why you played 12.Bxf6, giving up a Bishop, probably just to play the slightly-aggressive looking 13.Ne4 that achieved nothing. You could have just castled (on either side) put your B on f3 and moved your Rooks to the e-file. The resulting position has very little tension and would be completely equal. By playing 5.Qe2 you gave up on the possibility of building an early initiative.
Even after 12. Bxf6 your position is OK, perhaps a little worse, but 16.h4 is truly a terrible move. After this, you are a Pawn down, you have two weak Pawns, zero compensation and you have opened up the position to your opponents Bishops. You try to shrug this off by saying, in effect, "so what, its just a Pawn", but this is NOT a position where such nonchalance can be justified. 16. Bf3 would have kept the position roughly equal, neutralizing one of those Bishops. The Pawns would be completely balanced (not a passed Pawn in sight) and you have no difficulty mobilizing your Rooks, so it should not be hard to get a draw, but after your previous play a draw should have been the limit of your ambition. You had no right to expect more and the move h4 should never have entered your head.
ADDITION In response to a request from @Mast I quote from an online review from chess.com. You can find this and other reviews via Google.
A large part of Art Of Attack is devoted to what Vukovic considers to be the preconditions necessary for an attack against the castled king to proceed. Among these are the attacker's control of the center, prevention of counterplay on the side of the board opposite the opponent's castled king, proper posting of the attacker's pieces, and a weakening of the castled King's defenses. It is also important that the would-be attacker not commit to the attack until all the preconditions are met, and Vukovic writes at some length, with examples, about what happens when the attack begins too early or too late. Too early and the attack will fizzle, while if the attack begins to late, the attacker may have missed his window of opportunity.