It threatens e5+

32...e5+ 33.fxe5 Rxc5+!

Permanently blocking the e6 pawn. And probably the winning move.

Good move. Now black has to concede someting or the king makes it to b6

36.Nb2, Be8; 37.Nd3,Bf7; 38.Ne5,Be8; 39.g3, b6+ (forced?); 40.Kd4,Kd6; 41.h3, and black seems to be in some trouble. If he plays 41...Ke7 white can play either c5, or maybe even Nc6?!. And if he plays 41...,Kc7 then Nf3 causes black serious problems. A possible way out may be 41...,h6; 42.h4, Kc7; but after 43.Nd3,Kd6; 44.Nb2, Bd7; 45.Na4, Kc7; white seems to win again. So in conclusion, yes Nd1-b2-d3-e5 seems better.

I think one doesn't have to assume Black will simply lie down. White has to take into account the possibility that Black might (sooner, rather than later) try to mobilize his K-side by ...h6 and ...g5.

Weakening g6, I think he should have avoided this move.

Ordinarily I would agree, but it seems to me that this attempt to mobilize the pawns (which incidentally would free his bishop) is probably Black's best chance.

Black had to have taken this move into account. The problem for White from here on is that a minority attack (to loosen up Black's K-side pawns) is out of the question.

This could be a mistake, since white will be able to exchange c4 against b6 later. Then the remaining b pawn might become very strong. Its hard to calculate however. Instead of b6+ I might have tried Be8, since white playing b6+ does not look very dangerous.

I think this check had to be on Black's agenda somewhere. Note that it blocks off all entry into Black's position on the Q-side to the White King. However, White is threatening to nail the BK to defending this pawn (Na4), whereupon the WK invades on the other wing.

A possible try seems to be 38...g5 39.hxg5 hxg5 40.fxg5 Kd6 and now if 41.Na4, e5ch 42.Kd3 Kc7 with problematical winning chances for White.

This shows that b6+ was not blacks best move.

Well played by black