Actually, now win for white after move 12 is quite easy.
White should firstly play Kg6, that cover checking square with rook,
and then advance pawns to 7th rank.
It is well known that it is impossible to prevent 2 7th rank pawns
from promoting. You got technique right.
Also sometimes with inaccurate play QvR is possible,
but even this one is still simply won.
Even though I am now winning most of these endgame vs the engine, it is often due to a lack of consequence in its defense. After the moves 13. Kg6 Rb8 14. Kf6 Rf8+ 15. Ke6 Rf1 16.g6 Re1+ which execute just what you said (Kg6 and advancing the pawns), you get this position:
Apart from things being well known and the engine telling me that it´s winning, what concrete plan do you propose to win this position and escape the checks? After for example a little zigzag king walk
17.Kf6 Rf1+ 18.Kg5 Rg1+ 19.Kf5 Rf1+ 20.Ke6 Re1+ 21.Kd6 Rd1+ 22.Kc5 Rc1+ 23.Kd4 Rd1+ 24.Kc3 Rc1+ 25.Kd2 Rc8 26.Kd3
it is not so clear, is it?
I have wondered about this too, but I think I get it now. The plan is to get your rook to the f file.
Conctretely, you go 17. Kd7 Rd1+ 18. Ke8. Now black loses to Rg7+ if Rg1 or Rh1, so 18... Ra1 seems necessary. Then the key move is 19. Rf7.
Now after an avalanche of checks, white's king goes back to g5. So 19... Ra8+ 20. Ke7 Ra7+ 21. Kf6 Ra6+ 22. Kg5 Ra5+ 23. Rf5.
Now if black retreats to the first rank, white simply plays h7+, Kh6 and wins because there is no stalemate trick. On any other rook move, white queens. Say 23... Ra1. Then 24. h7+ Kg7 25. Rf7+ Kh8 26. Rf8+ Kg7 27. h8Q+. I think that does it but I may have missed something.
I made a mistake. After 19.Rf7 black has 19... Re1+ and white is not making progress.
But I still think Rf6 is the key idea. After your 25... Rc8, white plays 26. Rf6. Now there are three ideas. Black cannot defend against all of them.
Idea 1: white plays g7 followed by Rf8.
Idea 2: white gets the king to g5 and plays h7+, Kh6 and g7 mate
Idea 3: white plays h7+ Kg7 Rf8! and queens on h8.
After Rf6, black has three defences. Either give lateral checks, say on the a-file, or play 26... Re8 to cut the king off. On 26... Re8, white plays 27. g7 (Idea 2) and black's rook is tied to the first rank. Now white gets the king to g6 and wins, or else black will play Kh7, in which case white has Rf8 and will win once the checks run out.
If black tries lateral checks, say with 26... Ra8, then white runs the king to g5. For example, 27. Ke3 Ra3+ 28. Kf4 Ra4+ 29. Kg5 Ra5+ 30. Rf5. Now black either goes back to the first rank or at some point tries vertical checks. If black attempts vertical checks, then white wins with Idea 2 or Idea 3. If black goes back to the first rank, say 30... Ra8, white plays 31. h7+ Kg7 32. Rf7+ Kh8 33. Kh6, forcing mate as there is no stalemate trap.
After Rf6, black's third idea is to give vertical checks, say starting with Rd8+, but this just drives the king to g5 with the same result as before.
So to summarise, the winning plan is:
1. Use the king to advance the pawns to g6 and h6
2. Put the rook on f6
3. If black's rook stays on the 8th rank, move the king to g5, play h7+ and force checkmate in the corner.
4. If black's rook breaks out and leaves the 8th rank, play either g7 and Rf8 or h7+ and Rf8 to force promotion. This is where you can end up in a Q v R ending if you misplay the Rf8, h7+ idea.
So in Silmans famous endgame course he covers the endgame rook + 2 connected pawns (g and h) vs rook. This is the starting position and it is white to move and obviously win.