1.Qg6 hxg6 2.Nh4

That's very nice, both the mate threats and White having to lose a move because of Qd1-a4.

1.Kc6 Ka5 2.Kxc5 d2 3.Rd4 (3.Rc2? with the problem noted above) e3 4.Rd6! Ka4 5.Kc4 should be the main line.

1. Rf6+ Ke5 2. Bc1 (if Bd2 then Kd4 probably) Qb8 (seems like the only move to not lose a queen?) 3. Rd6! seals the deal, since after Kf5 white has e4 and after 3.. Qf8 4. Bb2+ Kd6 5. Ba3+ and Bxf8.

If I am correct it's frickin beautiful!

I'm too lazy to re-analyse it thoroughly right now, but I'm sure 1. Qg6 hxg6 2. Nh4 is met by 2... Rf5 closing the diagonal, after 3. Nxg6+ Kh7 the g6 knight becomes en prise if 4. Ng5+ is played, and if 4. Nf6+ gxf6, then 5. Nxe7 is with no check, so the smallest of evils for White is then the sad 4. Nxe7 Nxe7 5. Ng5+ remaining about two pieces down. However, you seem to have grasped the idea right

With the Black pawn on h2 and White King on g2 and the Black knight on f1/f3/g4, this is a known draw ( with the Black king away ); thus,

1. Bg1 Nf3 2. Bh2 Nxh2 3. Ke2= Ng4 4. Kf3 and the threat of Kg3 forces the Black pawn to the h2 square.

rats, i was hoping to catch someone with 1.Bg1 Nf3 2.Ke2 Nxg1 3.Kf2 Ne2 (fell for that myself in tactics trainer)

That was exactly my line, and I almost posted it as "This is what I've come up with so far, but I can't quite figure out if it works or not - and I kind of think maybe it doesn't". Then I gave up on the problem instead. Which is probably for the best because my tentative line continued 3. ... Ne2 4. Kf3, so apparently I have some work to do on my visualization skills, lol.

1. Rb7+ Kc6 {1. ... Kxb7 or Kc8 2. Nd6+ wins} 2. Nd8+ Kd5 {2. ... Kc5 3. Rc7+ wins} all seems nice and forced, and "feels" like the beginning of the solution, thematically, but I can't see where it would go from there... maybe there's a different idea entirely? Or maybe I just can't find the finish?

Woooow!

so 1. Rb7+ Kc6 2. Nd8+ Kd5 3. Rd7+ Kxe5 (3.. Ke4 4. d3) 4. f4+ and now 4.. kxf4 5. Rd4+ or 4... Kf6 5. Rf7 wins easily, but it took me a while to see what to do after 4.. Qxf4. 5. Rd4!!! Seems to be winning Forks, forks everywhere!

I guess all of the Queen endings that aren't obviously winning are drawn?

Without moving the pieces, being unrated (2050 on Chesstempo), so probably missing a ton or even giving completely wrong answers, but after 40ish minutes:
1. fxg6+ Kxh6 2. g7 (Qg5+ Kg7) Rxf4+ (Qb5+ Be2)(R8 Qg5+ Kh7 Bc2+) 3. Qxf4 Kxg7 4. Qg5+ Kf8 (Kh7 Bc2+ Kh8 Qg6+ Kg8 Bh7+ etc, Kh8 is met with Qh5+) 5. Bh5 Qxy 6. Qh6+ Kg8 7. Bf7#
1. .. Qxg6 2. Bc2 Rxf4+ (Qxc2 is met with Qg7#) 3. Ke1 (g-file allows Rf6 Bxg6 Rxg6 with a pin, e2 allows Qxc2+)
1. .. Kh8 2. g7+ Kh7 (Kg8 gxf8=Q++ Kxf8 Qg7#) 3. Bc2+ Kxh6 (Kg8 h7#) 4. Qg5#

The reply I'm most unsure about:
1. fxg6+ Kg8 2. g7 Rf6 3. h7+ Kxh7 4. Bc2+ Kg8 (forced) 5. Qh4 Rxf4+ (5. .. Kxg7 6. Qxg7 mating|5. .. Qb5+ Kg1) 6. Qxg4 I think should win.

Earlier try:
1. fxg6+ Kg8 2. h7+ Kh8 3. g7+ Kxh7 4. gxf8=N+ Qxf8 5. Qh4+ Kgx 6. Qg5+ Kf8, or else Qh5, 7. Bh5 Qxy 8. Qh6+ Kg8 9. Bf7#
2. .. Kg7 h8=Q+ Rxh8 (Kxh8 g7+ Kg8 gxf8=Q+ Kxf8 Qg5 and I don't think Black can defend) 3. ??

Edit:
1. fxg6+ Qxg6 2. Bc2 Rxf4+ 3. Ke1 allows Re4+ breaking the pin :\ Maybe going to the g-file and allows a trade of pieces after Rf6 leads to a winning endgame? King seems to be fast enough.

All I can really say is that I'm instinctively drawn toward 1. Bh5, but I have nothing concrete, assuming my opponent isn't foolish enough to just take the bishop. Probably 1. ... Qb5+ and then who knows what's happening?

And I'm only posting this to log my thoughts at the time I gave up, for comparison to the actual line when.

1. fxg6+ Qxg6 2. Bc2 Rxf4+ 3. Kg1 Rf6 4. Bxg6+ Kxg6 5. Qxg6 Kxg6 6. Kf2 Kxh6 7. Ke3 leading to a winning pawn ending.

Did I miss something or did I get lucky?

EDIT: Okay, I see this pawn ending is drawn

there is a very beautiful win in one line: 1. fxg6+ Qxg6 2. Bc2 Rxf4+ 3. Kg1 Rf6 4. Qg5 b5? 5. Bd3! Rxe6 6. Kh1! Re1+ 7. Kg2 Re6 8. Bb1! Rf6 9. Kg3 e5 (Kd6 Bxg6+ Rxg6 Kf4) and now the pawn ending is winning

but I think 4...Rxe6! refutes the whole thing. If White could reach this position with his pawn on b5 he would win by escaping the checks with Kf1, Ke1, Kd1, Kc1 - but with pawn b4 Black plays 8...Rc6 and draws. Not sure if this is going to be the theme that wins it or just a really nice red herring. But then White must have gone wrong earlier, maybe even on move 1?

I hope this is thematic and the solution looks a little like this, I will be very disappointed if the final position is just a favourable Q and 3 vs Q and 2