Bf4 is better

White has accepted the exchange of his dsb from the moment he played it there...

So, white lost his potential attack on the e column, but has a strong center, his dsb can get into the game, and the Nb1 balances the Bc8, while the slight weakness on the kingside seems far to be exploitable by black. I wouldn't say black played the best of the opening, but I know I'm far from their ratings...  

This is a new start of a game...  

to what end?

This moves precludes c4 to both B and N, and prevents Qa4, b5... black has no attack and has to control white's newly remade army before it becomes dangerous. The only other way black could have taken would have been e5. But white has not to take, he can easily hold with e4 in return. By the b5 way, black has also the b7 strong square for his lsb!

a weak move, 14.a4 attacking the overextended pawn chain is better.

A fantastic idea, that proves why you are where you are! Bf1 now PLAYS from his weird square! Now, looking at it, it seems that black's best answer to 14. a4 is b4, and white can't take or Nxb4 --- Now, back to the game: white of course is looking to the missing Nf6 and sees all the field free for his tho-pieces battery. Maybe a little "naive"?

Preparing a powerful attack on g2. Black is already thinking in Bh4, but there is nothing concrete after NxB, QxN, Nd2-f3. Patience. a3 by white, so to prepare c4 can be another plan, and Qc2 could have be better that the lsb move.

I'd assume White wants to contest Black's strongly-placed bishop is what he was thinking?

15.Qe2 or 15.Ne4, both deserved consideration

It's always a bad idea to put a piece from where it can be harassed. Now that black has castled, he can do f5, and the threat to e6 is still far to become a reality. True is black has no urgent need to to that, so why waste the move?

I would preferred 15..f5 here, calling white's bluff on e4

Black's plan has nothing to do with h4, then. I didn't see much that way either. Here, the hidden W weakness is e3.

Black is better off because he has the 2 B's, and white has weakened several squares near his king

In sight of the answer of white at the more-than-natural Bh4, one may wonder whether W had "seen" his next move after Bh4 or not. The N move opens the way to another W piece...

Joe you should have added this variation too 17. Bxh7ch KxB 18 Qh5ch Kg8 19 QxB, and you should be able to work out what happens if black 17..Kh8

As you can see, the move order DOES make the difference: kenneth_fiore_3 was right: the line starting with Bxh7+ was the ONLY one: now black has one "saving" move:

A juicy position. Observe that black will lose a piece by force because the WB has a "forcing escape route": BxN (BxB, QxB and white won a piece). BUT this doesn't work THIS time because after BxNc6, Qxc6 threatens mate on g2 (d'yuremember?  . White doesn't win a piece, just a pawn after QxB, fxB, N (or Q?) xe4. BUT, in sight of that, WHAT IF white plays an in-between move, like 18. d5 or 18. dxc5, in order to complicate things with another threat... ? It's a tactic mess I found fascinating, but my conclusion is that it does not deserve anything better to white that the exchange. 18. dxc5 forces a Q's movement by black, and he has Qxc5 or Qd8. (The idea is to pin the Q's, but it's not good, because of 18... Qd8!, BxNc6, black can choose BxBc6 and white has two threats -BxR and fxB to meet). 2) 18. d5 is also interesting: black can't take or Bxd5+ and he will save the piece. If BxR instead, this crazy exchange chain can go on like this: 18. d5, BxR, dxN (if black Bxc6, BxB and white "saves" the B: black "must" take it or BxR, so QxB, RxBe1. White will lose the exchange, but the position is far from even. But it's crazy to go on: 18. e5, Black seizes the R: BxRe1. If white takes it, no good after fxB, so 19. dxNc6. The Q can't take it, and if Bxc6, BxB and white solves a major problem. But if black wants to go on: BxNd2, 20. cxBb7, this time threatening a R. Look at the board: does have black "something" or not, here?

In other words: white, instead of winning a clean piece, got to win a central pawn and an improvement of his position, adding a weak e6. A disadvantage for black, but still a complicate position! I love these games!

c4, with the idea of b4 soon after...?

AND e6, heavily. The weakening effect of f5 is now felt in all its strenght.