What went wrong?

Can anyone explain what went wrong in the endgame and why it was so wrong?
[Event "Rated Rapid game"]
[Site "lichess.org/QkceOmGS"]
[Date "2020.06.03"]
[Round "-"]
[White "Yazan288"]
[Black "cristof012"]
[Result "1-0"]
[UTCDate "2020.06.03"]
[UTCTime "10:58:28"]
[WhiteElo "1976"]
[BlackElo "2012"]
[WhiteRatingDiff "+72"]
[BlackRatingDiff "-6"]
[Variant "Standard"]
[TimeControl "900+10"]
[ECO "D53"]
[Opening "Queen's Gambit Declined: 4. Bg5 Be7"]
[Termination "Normal"]
[Annotator "lichess.org"]

1. c4 Nf6 2. d4 e6 3. Nc3 d5 4. Bg5 Be7 { D53 Queen's Gambit Declined: 4. Bg5 Be7 } 5. cxd5 Nxd5 6. Bxe7 Qxe7 7. Qb3 Nxc3 8. Qxc3 O-O 9. Rc1 Nc6 10. Nf3 Bd7 11. e3 Nb4 12. Ne5 Nxa2 13. Qc5 Qxc5 14. Rxc5 Ba4 15. Bc4 Nb4 16. b3 Bc6 17. Nxc6 Nxc6 18. Bb5 Nb4 19. Rxc7 Rac8 20. Rc4 a6 21. Kd2 Nd5 22. Rxc8 Rxc8 23. Be2 g6 24. Ra1 f5 25. Ra5 Kf7 26. Bf3 Rd8 27. Rc5 Rd7 28. Bxd5 exd5 29. h4 h5 30. b4 Ke6 31. Kd3 b5 32. f3 Rd6 33. e4 dxe4+ 34. fxe4 fxe4+ 35. Kxe4 Kf6 36. d5 Ke7 37. Rc7+ Kf6 38. Kd4 Kf5 39. Kc5 Rf6 40. d6 Kg4 41. d7 { Black resigns. } 1-0

29...h5 leaves g6 permanently weak
31...b5 leaves a6 permanently weak
35...Kf6 36...Ke7 37...Kf6 cannot be right, but it already was too late

I think a golden rule when defending slightly worse endings is that you should be comfortable doing nothing. Let your opponent try to find progress and only weaken yourself for counterplay when absolutely necessary. I think objectively you are fine until 29. ..h5?, which is completely unprovoked. You prevent your opponent from going h5 but so what? If he opens the h-file that is more counterplay for you. After 31. ..b5?? I think your position is probably not humanly possible to hold without a lot of time. Again, you didn't need to lash out, just play the position is-is:

Here's how you hold the position at move 31: You play Rd6 and wait. First of all, black cannot penetrate via Rc7 or Rc8, because you go Rb6 Kc3 Rc6+, trading into a drawn pawn-ending. If the white king steps to the kingside to threaten Rc7-stuff (for example directly Kc3), you can play f4 exf4 Kf5 with enough counterplay, because you clean up the entire kingside. Again Rc7 or Rc8, which looks scary, fails to Rc6+ where white has to be careful not to lose actually. Otherwise you can move your king back and forth until white tries something, for example if f3 then you move your king to e6 and get Rc6 going, also trades into a drawn pawn-ending, as long as you calculate your tempi correctly.

I cannot give you a full variation to a dead draw, because white can play this position forever, but just know that you are fine as long as you just sit tight and always have a strategy on what to do when white tries to make progress. But as long as he doesn't make any progress, you don't have to do anything other than move back and forth.