The following position arose after 19. Qf3:

I agonized over 19. ... gxf4 or 19. ... Bxf4. Stockfish rates gxf4 slightly better (so slight that I don't think the difference matters), and that's what I chose, but my reasoning for doing so was flawed. I was worried about the queen being trapped on h4 after Bxf4, even though 20. Bxf4 gxf4 would likely follow. How should I have approached thinking about this position? Also, I welcome comments on the rest of the game, which was very fun up until the blunder at 27. Rxf7.

[Event "Playing on Chess Time"]
[Site "ChessTime"]
[Date "2015.03.11"]
[Round "1"]
[White "ShortStump"]
[Black "STinLA"]
[Result "0-1"]

1. e3 e5
2. Bc4 Nf6
3. Nf3 d6
4. O-O Be7
5. Ng5 d5
6. Bb5+ c6
7. Bd3 e4
8. Be2 O-O
9. d3 Bf5
10. g4 Bg6
11. Nc3 h6
12. Nh3 Nh7
13. f3 exf3
14. Bxf3 Bd6
15. e4 dxe4
16. Nxe4 Bxe4
17. Bxe4 Qh4
18. Nf4 g5
19. Qf3 gxf4
20. a3 Ng5
21. Qg2 Nxe4
22. dxe4 Nd7
23. Bxf4 Bxf4
24. Rxf4 Ra8d8
25. Ra1f1 Nb6
26. e5 Nd5
27. Rxf7 Rxf7
28. Re1 Qxe1+
29. Qf1 Qxf1# 0-1

Just reviewed this by importing into lichess.org (first time I've done it). What a great tool! I didn't even realize during the game that I missed Ne5 two moves in a row (24 and 25), and that he missed g5 two moves in a row (25 and 26). Basically, it was a battle of us trying to gift the game to each other.

God, I'm horrible. Not just for making the mistake, but for not even realizing it.

Well, 19.-gxf4 20.Bxf4 Ng5 seems annoying for White, so I'd go for that just to reduce his options/give a chance to go wrong (I'd expect 19.-gxf4 20.Bxh7+ Kxh7 21.Bxf4 Bxf4 22.Qxf4 and now probably just Na6 (not Nd7 Qf5+ forking)
It's good that you blunderchecked the game, make sure you understand what were the ideas behind evaluation swings.
Not playing 2.-d5 or 3.-d5 is a pretty fundamental opening play flaw, d5 achieves so many things (central control, tempo, closing his bishop's diagonal).

Thanks. I was just using first level thinking of attacking the center, but didn't consider limiting one of the natural squares for him to develop the king's bishop with d5.

This guy likes playing things like 1. c4 or 1. d4 ... 2. e3 ... 3. f4 that you don't see a lot at my level. I like it because I can usually play Nf6, Ne4 which ends up being horrible for him as things develop.

Ideal thought process to work towards: "d5 looks right, is it any good?" (At this point, you'd notice the good things it does.)

Pointing this out because your phrasing suggests that you'd want to derive that d5 is good by trying to accomplish various objectives and then noticing that d5 is compatible with them. That would be too unreliable and time-consuming.

d5 establishes an ideal center, so it's a sort of priyome in itself, to be recognized first and then evaluated second. I wonder if there are pattern drills for this other than "see it in a variety of contexts, scattered over the course of many months".

I don't intuitively grasp why d5 creates an "ideal center." What are the qualities of an ideal center that favor d5 over e5? Is it because d5 would control a center square he doesn't already control?

The arrangement of d/e pawns on the fourth rank is named the "ideal center". That's chess terminology, not casual use of the adjective "ideal". The move d5 would have brought the ideal center into existence.

Generally you should arrange the pawns that way if you're allowed to; they control important squares / allow fluid development / etc.

In fact, you'll notice that in many standard openings, white plays e4 and d4 unless black prevents it (or if white goes for d4 and c4 instead, but that's a slightly different story).

Understood. Thank you.