If you click onto the Lichess analysis board linked, the second button from the left on the bottom-right panel lets you play the position against the computer. Try it and see if you can actually prevent the pawn promotion that way...

As far as I can tell, that wouldn't work. White king couldn't make it there in time to prevent a black pawn from making it to the other side of the board and getting promoted, not if the black king also moved toward in to block. Once black king prevents white king from blocking the pawns, it's just a matter of time.

It might be that the key here is to get a head start on blocking those pawns. The first move should be to move white king to the left, putting it in a position better suited to do that.

I'm not sure. Black king rushing to 2-row could stop white from reaching C1 by moving to B2. Without occupying C1, white cannot stalemate.

Now what if white plays King to H1. Black queen cannot capture because that's a stalemate, and cannot get out of the way because it's a check. But I'm not sure how the game goes from there.

I think you’re right. Looks counter-intuitive because trading queens puts the black king in the perfect position to capture our pawn, giving black two passed pawns against a lone king. Didn’t calculate it all the way through, so I don’t know whether it’s black or white who’s going to be stalemated.