Here are a few of my thoughts after playing through the game. I'll start with the position after 31...e5 since you asked specifically about the endgame. One thing to keep in mind is that in positions like this, where you have an overwhelming material advantage, there are usually several ways to win. The moves I'm suggesting might not be the best, but they're the ideas that come to my mind first and should be sufficient to win.
-I like 32.fxe5. There's no need to allow black to push the pawn and get a passed pawn, which is a tiny hint of counterplay (even though white is still very much winning). Simpler would be to trade the pawn and take away black's only play. Something like 32.fxe5 Rxe5 33.Rf4+ Ke6 34.Rf2 looks good. If black trades rooks, white is happy because two rooks are much more dangerous to a king than one. If black doesn't trade, then we simply put the rook behind our pawns and we're off to the races.
-38.Rg7+ looks better. After 38.Rg7+ Kf6 39.Rxa7, white wins the a7 pawn and is about to win the e2 pawn as well.
-43.Rxa3 Kxg7 44.Rb3 looks like an improvement as well. Again, trading a pair of rooks is good because black has no more mate threats. And from b3 the rook protects the g3 pawn and prepares to push the b-pawn. The rook is in an ideal position and there's nothing black can do.
I hope some of that helps. I don't like it when chess authors give just variations, so I'll try to give some summary principles as well. As usual in chess, there are not any 100%, always the case, type rules, but these should be pretty standard.
1) When you're up a significant amount of material (a rook, or two minors, or 5 pawns), it's best to trade down any chance you get. This will highlight your advantage and leave you with the only winning chances.
2) If you're on the good side of a 4-rook endgame (side with more pawns and better winning chances) do all you can to get a pair of rooks off the board. As happened in your game, the defending side can create mating threads or annoying perpetual checks. Conversely, as the defender, whenever possible keep all rooks on the board and be as stubborn as possible.
I hope some of that helps a little bit. Feel free to ask more questions and I'll be happy to answer as best I can. Keep in mind I'm a patzer too