Name for games like this? How do you break through?

Is there a name for games like this?

It's like the whole game is a fortress, but it's not really. The computer analysis says I was doing very well throughout, and my opponent started just making random waiting moves at the end due to the time situation, but otherwise I wasn't super sure how to break though. Is there a clear strategy for black at turn 33ish? I felt that the f3 pawn was the weakest/easiest to attack, since I didn't have my own pawn in the way, just like on the c-file, but it also had no pawns defending it. I also felt like having my e3 pawn meant it was hard for him/her to defend it, as well as having a nice white diagonal (with my white square bishop).

Conversely, it doesn't seem like I have any bad pawns at all. How can white even get counterplay? (Also s/he has so little room compared to me)

Yes, the plan is attacking the backward pawns, which you did. Your biggest mistake in the closed position was pushing h4, it is better to keep the pawn tension because you have the plan of doubling up rooks on the h file and then taking. White doesn't want to take to give you the half open h file as you have too much space and pressure.

I kept trying to close the game up, since I wanted more space, and I felt like my kind was really hard to defend with every pawn pushed away. Good point though on whether to push h4 though.

More space doesn't mean much though (if there's no way to break through). And if he just left his N on e4, it was gonna be hard to pick on the f3-pawn.

Well, is only one pawn break (a4) so that is the obvious first try to break through.

For instance on Move 29, ...Bd7 > ...a4 seems like a simple plan, no?

You did miss the en passant capture bxa3 on move 33.

Queen recaptures and Black has nothing though?