Is it worth ruining his castle with this exchange?

1.Bxg8 fxg3 You just lose tempo. 2.Bishop is developed but Knight is not developed You lose tempo. 3.Bishop already prevents white king from castling. keep your tension. etc...

fxg3 > fxg4

What move are you talking about?

Your light square bishop was hanging. They are saying white should have played 11.fxg4, keeping the ability to castle and forcing the dark square bishop to retreat.

NO IDEA what you're talking about.  fxG3 is not even a valid move for white.  Do they even understand what the greater than symbol even means?  WTF

He means that after 10 ...Bxg1, white can play 11 fxg4. Then you'll have to move the bishop to avoid losing a piece. Then white can castle kingside or queenside.