I think the waiting move is the best here. Can't really say if it's Qh4 or Qh5, or some another. It's a losing position unless an opponent makes a mistake.
Funny position.

#4 actually not, it's pretty simple. If you find the right first move, it's all a simple case of Zugzwang, the black bishop has to guard the g2 pawn, the Knight on h8 has the guard the h8 square so white can't promote. And remember, the black pawns are also up for promotion

the moves are qc3, kh2(black) ,qxh8, kg3(black) ,qg8+,kh4(black) and h8=q#
Kg3 isn't forced for black, and with two new queens for black, black actually wins
Remember to always see your opponents' plans too!

Nice idea.
White's useful moves which don't harm something are the king moves on dark squares, while Black has to move the h8 knight to f7/g6 and then back.
White cannot touch this knight with the king without losing a tempo, and the only way to achieve that is using the secure a7-a8-b8 triangle.
So: 1. Kb2 Ng6 2. Ka3 Nh8 3. Kb4 Nf7 4. Ka5 Nh8 5. Kb6 Nf7 6. Ka7 Nh8 7. Ka8 Ng6 8. Kb8 Nh8 9. Kc7 Nf7 10. Kb6 Nh8 11. Kc5 Nf7 12. Kd4 Nh8 13. Ke5 Nf7+ 14. Kf6 and white is winning.
Obviously the moves by both sides are not unique.
I also don't buy the Stockfish stuff. While it's quite possible that the crippled version of Stockfish which is used at this site would probably fail to solve it (or take a very long time), it will be an easy job for any local stockfish on PC, tablet or even a modern smartphone. And a "matefinder" type of engine like TheHuntsman 1 would likely find mate in NN moves in less than ten seconds.
Try to win with White. It's actually pretty funny.
(I just have the position here because I think it'd be pretty funny to read your ideas and start a really big discussion between <600s, Normal Players and Stockfishers. Btw, Stockfish isn't able to solve this)
The Position:
Have fun solving!