11 Nh2 is also a better move than g3. Just let him capture your bishop, followed by g3, and you will still be ahead by one pawn after Qxg4.
Is advancing the pawn to h3 after castling king-side ever useful?

11 Nh2 is also a better move than g3. Just let him capture your bishop, followed by g3, and you will still be ahead by one pawn after Qxg4.

"Is there any point in advancing the pawn to h3 after castling?"
Generally speaking, the Strongest(safest) pawn formation while castled is a flat pawn structure.
h3 on move 7 was not needed. As you said, the follow-up moves were not a threat or rather pretty slow. Even the Move g5 can't be done easily.
His goal here is to open a file onto your castle. Just prevent that or delay it until you have an attack on his King.
So I would have somewhat ignored his advances and delevelop. bNd2 as your queenside is not fully developed.

Is h3 EVER USEFUL after castling Kingside? Yes. Sometimes the small pawn weakening move is useful for prophylaxis, or a King luft. Is h3 USUALLY a good idea after castling Kingside? No.
As @Chingus_Khan mentioned, "[generally]...the Strongest(safest) pawn formation while castled is a flat pawn structure." From the perspective of pawn structure, pushing pawns from this "flat structure" creates pawn/square weaknesses - this is especially dangerous in front of your castled King. The concepts of square weaknesses can be positionally complex for most chess players (probably 1800+ rating chess may begin to better understand its implications with maybe 1600-ish rating being introduced to the ideas of square weaknesses more formally), but all you have to know it that GENERALLY you want to keep your pawns unmoved in front of your castled King, but that there are exceptions in some cases.
In the position posted, I don't like h3 because it weakens the Kingside (positionally due to weak squares) and because h3 doesn't give the Bishop a retreat like in my London System example (my second diagram), since here, Black just takes the h2 Bishop.
As others have said, it depends. h3 provides a luft (escape square) for the king if the back rank is weak, denies your opponent access to g4
sorry. and denies access to g4 if this is going to make your position uncomfortable. However, it also weakens the pawn protection around the king (if castled kingside) by creating a target for a pawn storm, piece sacrifice, and makes other moves such as g3 more difficult should they be necessary (or provoked). As with all things, if h3 solves an immediate threat that cannot be dealt with in some other way then play it. Try not to play h3 automatically (ghosts) or if another defense exists.
It is usually a bad idea to castle before your opponent does. The e3 pawn actually makes it hard to get defenders over to the kingside, so your opponent simply launches into an attack, not bothering to castle himself (though he could have castled queenside and added the other rook to the attack).
In general, moving pawns in front of the castled king will tend to weaken your kingside defense, as it leaves you more vulnerable to a pawn storm. Sometimes it will be necessary for tactical reasons, but generally you shouldn't play a move like h3 unless you really have to.

"Whatever I do next will be met by QH1#" actually Bb5+ or Be4 stops the mate threat and then allows you to get rid of the knight. Be4 is obviously better.
the mate threat also wasn't unstoppable. 11.Qxg4 leaves you with an albeit small edge.

Ok, I hadn't seen those possibilities to escape, but since both moves lose the bishop I don't see how Be4 is "obviously" better than Bb5+.
As for 11.Qxg4, wouldn't that be met by 11...h5 then either 12.Qg2 (to stay near that open h file) exf4 13.Qf3 (only safe square), and then back to what happened in the match, or 12.f5 e4 13.Bb5 (or Be2) Bxf5 forcing the queen to move away and then again back to what happened in the match?

Ok, I hadn't seen those possibilities to escape, but since both moves lose the bishop I don't see how Be4 is "obviously" better than Bb5+.
As for 11.Qxg4, wouldn't that be met by 11...h5 then either 12.Qg2 (to stay near that open h file) exf4 13.Qf3 (only safe square), and then back to what happened in the match, or 12.f5 e4 13.Bb5 (or Be2) Bxf5 forcing the queen to move away and then again back to what happened in the match?
Bb5+ let's Nxg5 take the bishop and so just loses the bishop. Be5 dxe5 Qxd4 swaps a bishop for a knight.
11. ...h5 is impossible because there's no h pawn after move 8 (which is before move 11).
May I suggest using the analysis feature on chess.com (htat you can find by clicking the bottom left of the board you posted. This give you access to an engine that will evaluate the position, saying who has the advantage and suggesting best lines.
I had already made this topic, but it seems to have disappeared for some reason...
On move 7 my opponent played a kind of gambit, sacrificing their knight for an unstoppable attack on my castled king. It's obvious in retrospect that the knight wasn't posing any kind of threat and I should just have left it alone, but that begs the question: Is there any point in advancing the pawn to h3 after castling, since it doesn't prevent the knight from going to g4? Or is it only a bad move after the black pawn moves to h5?