When is it fine to push away the enemy bishop with kingside pawns?

Ignoring all the obvious mistakes I made there's something I don't understand. On the analysis board stockfish suggests to play 8. g5 but in a lot of opening videos I've seen it's not played nor suggested because it weakens the kingside, both as black and white. I've also heard it's possible but only when you haven't castled yet, why? And for example in that game wouldn't castling queenside not be a viable option because there's no c4 pawn?

g4 when castling
stockfish that cheating bastard had all the positions up to 22ply already on the board so it can see that there is no mate. For a human player this is to complex because we are not able to see. To complex to calculate if you get mated = bad move.

g4 opening.
in some opening lines you play g4 quite early this is all preparation.

Castle with missing c pawn.
Why do you castle: Save king and connecting rooks. Without the queens on the board it is very hard to mate.

also
You can always move to the b file and take the c file for your rook.
in lots of openings the first move after 0-0-0 is Kb1. Or if you do not know what to play play Kb1.

it comes down to seeing threats. and g4 is a perfect solution to this position. it weakens ur king side but wins the bishop and you have a nice pawn formation. other option is to play Nf3. So really the positional feel is something u develope over time and all chess players work on getting better at it.

Side note. whenever u play 4. e3 You have to be aware that you are gonna have hard time to get ur bishop into the game so better option would be to play nf3 because it develops a piece and doesnt block your bishop.

8. Bd3 also a bad positional move, If ur pinned u usually wanna kick away the piece that is pinning you or block the pin and it also blocks the queens support to the d4 pawn. So a move like Be2 would be better or even Bb5 pinning him back. but these are just minor mistakes that u will slowly start to understand.

@fridayfreak Thanks for the tips. Why do you say it wins the bishop though? Do you mean I can exchange it for my knight afterwards?

And yeah I know that about 4. e3. The game before that was against the same guy and opening where I played 4. Nf3 instead exactly because of that, but I played it poorly and thought e3 was better because the post match analysis board suggested me to play it. I'm still not sure if I should try my best to follow engine moves or not.

I 100% agree 8. Bd3 was a very poor move, I don't remember why I even played it. I had my doubts before making that move but still did it.

Steinitz and others have advised to kick a bishop on g4 immediately with h3.
In the above example 7 d5! was even stronger, as 7...Ne5 fails to 8 Qa4+.
8 Bd3 is also weaker than 8 g4! intending 8...Bh5 9 d5.
If you play both h3 and g4 it usually is with the intention of O-O-O and attack on the king's side with g5.
Kasparov won several games with 1 d4 d5 2 c4 e6 3 cxd5 exd5 and preparing O-O-O and attack with g4, even without black ...Bg4 h3 Bh5 g4. The white king usually goes to b1 for safety. So the move ...Bg4 wins two tempi for white.

It's somewhat difficult to ignore your other mistakes, as you requested, and answer only the question about pawns chasing the bishop, because the answer to the latter is related to the former. You are right in your remark that h3 followed by g4 weakens the kingside and should usually be avoided. I would say that when you have a fianchettoed bishop protecting your king (in this case it would have been the light squared bishop) it would be safer and more desirable under many circumstances to chase a bishop with h3 and possibly to follow it with g4.
In the current game, I think you were fine after moves 9, and could have played 10 Qa4+. If 10....c6, then 11 Nxe5. If 10 ....Ned7 then still 11 Ne5 or 11 0-0. Someone above said Bd3 blocks queen's protection of the d4 pawn. Only problem with that comment is: there is no d4 pawn to protect in this case. Bd3 is often a good place to develop because it attacks h7 (sometimes supporting mate) and also helps control the centre (for instance if you eventually advance the e pawn to e4). All that said, the enemy bishop is now on a dangerous diagonal, as the ending trap proved. So you have to play accurately and carefully. That is why the mistakes are related to the answer. You don't want to weaken the d file, which opened in the first moves, unless you can castle before it can be used to attack your king. On the other hand, you don't want to castle when you've moved your queen away, if your N is still on f3 and he can play Bxf3, forcing you to recapture bxc3, ruining your protection of the king. I believe 10 Qa4+ offers a solution without having to play h3 g4, but Be2 would have been safest.

In Sicilian dragon, chasing the white bishop from g5 with h6 then g5 is very strong, and recommended by stockfish. But as I said, black's dark bishop is fianchettoed, making it safer. (There's more to the story, of course). When you do chase a kingside bishop with pawns, it is often desirable to follow it with a N attack on his B if he keeps it on the kingside. That way you get a B for a N, and also eff up his kingside pawns a bit in return.

Thanks a lot for the advice, really helpful.

When is it "fine"?? What does that even mean?

@tim808 I mean when is it a good idea, because I've heard it weakens the kingside and is sometimes a bad idea