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Here I go:

1. Kg3 Rd8 -> What now?1. Kg3 Qg1+ -> What now?1. Kg3 h6 -> What now?1. Kg3 h5 -> What now?
2.. Qd5+ Rxd52.. Kh3 Qxh2+2.. Qxa8+ Kh72.. Qxa8+ Kh7
3.. a8=Q+ Qb83.. Qxh2 Qxa43.. g6+ Qxg6+3.. Qf3(attacking a5) Qg1+
4.. Qxb8+ Rd84.. Qg2 Qxh74.. Nxg6 Kg64.. Kh3 Qxg5
5.. Qxd8# easy xD5.. Qd5+ Kh85.. Bc2+ Kg55.. Bc2+ g6
6.. Ng6+ hxg66.. Qd8+ Kh56.. Qb7+ Kh6
7.. Rh4#7.. Qh4#7.. Nf7+ Kg7
8.. Nxg5+ Kh6
9.. Qh7+ Kxg5
10.. Qxg6+ Kxf4
11.. Qg3# (Qf5#) Very very difficult xD

After

  1. Kg3 Qg1+
  2. Kh3 Qxh2+
  3. Qxh2 Qxh4 is not possible. In case you mean Qxa4, then
  4. Qg2 Rd8 -> What now?

Interesting is also:

  1. Kg3 Qg1+
  2. Kh3 Qxh2+
  3. Qxh2 Qa3 -> What now?

After

  1. Kg3 h6
  2. Qxa8+ Kh7
  3. g6+ Qxg6+
  4. Nxg6 Qg1+ -> What now?

The last sequence (1. Kg3 h5) is accepted, but please check your notation again (4. Kg3 is not possible).

By the way, just as a hint: chess software is not allowed to solve my problems (apart from some exceptions where I explicitly inform the competitors). I am sure you are aware of that, are you? :-)
I mention that because your variations are very accurate without any small typical human errors ... I didn't say you are using chess software - just wanted to state the rules again, just in case ... :)

Edit: Sleeping time for me: I will check all remaining answers tomorrow. Good night for now. :)

  1. Apologies for the notation, with so many letters it's easy to get wrong
  2. I know that chess software are not allowed, I think moves are forced in most cases, anyway, I present you my chess software :)
    WhatsApp Image 2017-12-27 at 8.59.34 PM.jpeg
  • 4.. Qg2 Rd8 -> What now?
    5.. Rxd4!! This move took me a long time to see it
    If Rxd4, Qa8#
    If Qxd4, a8=Q winning the game.

  • 3.. Qxh2 Qa3 -> What now?
    4.. Qg2 (attacking the Rook again) Rd8
    5.. a8=Q Qd6
    6.. Bb2+ Kh8
    7.. Nf7+ Kg7
    8.. Nxd6+ Kh8
    9.. Qd8#

  • 4.. Nxg6 Qg1+ -> What now?
    5.. Qg2 Qe1+
    6.. Kh3 Bxa7
    7.. Rf8!!

I'm very sorry if I made you doubt. Chess is my favorite sport and I always try to do my best.

Don't worry! I had to ask that. It seems you are very talented & engaged solving my chess problems. :)
I will put an official solution now with a few selected variations.
I hope you had solving this chess riddle which cost me quite some time to compose. :)

tnx for resteem