To prevent backrank mate, which pawn should you push?

In general, I would say the position and opponent pieces on the board will determine which to push. In this case, you can probably push any of them or even move the king.

I agree. You're up a queen for two pawns, so any reasonable move wins. What you did trading in your queen for the rook is fine here because it leads to an easy win. In this type of position g6 gives your king a nice fortress, Kg7 and the king is protecting the f and h pawns. Or Kf8 followed by Re8 and white's rook will either have to be traded or vacate the open files since your queen covers c1 and d1. Bottom line there are many ways to win here, you just have to not blunder.

In general, playing h6 or g6 is common, depending on your opponents potential threats.

In your position your a whole queen up, so it really doesn't matter which pawn you push since there are no potential dangers. Kf8 is probably the easiest to prepare Re8.