"Preserving the Dragon Bishop"

Your first game I think that the big mistakes were dropping the knight and then allowing the mate at the end, both of which could have been prevented after you moved your bishop to h8.
In your second game, your kingside was under attack and already torn up, his dark bishop was participating in the attack, and you sacked your dragon bishop for a knight. Trading your dragon bishop for his same colored opponent is sometimes good, sometimes bad. But, trading it for anything else when queens are still on the board and he retains his dark squared bishop is often fatal, and generally not a good idea.
The mistakes I mentioned in the games you posted should be of more concern to you then whether or not you should move your bishop to h8 after the enemy bishop arrives at h6. I have wondered a bit about Bh8 myself though since I don't really see it in any of the master games I playthrough, but then usually white doesn't seem to attempt Qd2--Bh6 in the games I look at.
Yeah, I'd say trading your Dragon bishop for that knight in the second game wasn't the right route to go.
I feel like 11... Qa5 would have allowed a slightly better attack for you, conforming with common counterattacking strategies for Black in the Dragon, not to mention further protecting the h5 square, in case White feels like saccing his Rook for your Knight on h5 at some point, another tactic I've seen in the Dragon quite a bit. Pushing with that pawn can often lead to counterplay, but I feel a better idea would have been to try putting your rooks on the c-file, as is commonly seen, and just sac for that knight, and perhaps after bxR follow up with Bxc3, and you own that whole diagonal, allowing for some very nice attacks. I might be missing some analysis, but that was just my first impression.

I think it really depends on the game. In some cases, one wouldn't do anything, and not bother wasting time on the bishop in order to create your own threats. In other cases it would be justified to move the bishop, even if it means sacrificing the exchange. And even other times it would be best to take on h6, because it would force their queen to h6. You would force their queen to h6 if you could sac the exchange on c3 the next turn, and get good attacking chances by doubling their pawns and gaining more central control.
In this post, I want to discuss the topic of black preserving his dark-squared fiancettoed bishop -- sometimes called the "Dragon Bishop" -- in the Sicilian Dragon and King's Indian defenses, with the move Bh8. This situation can occur after white attempts to force the trade of this bishop -- usually black's strongest piece -- by attacking it with the move Bh6. In such configurations, white's attacking bishop is always supported by his queen, usually from d2.
Often, white will attempt this "attack on the dragon" while black's rook still sits on f8, leaving black with no choice but to trade the dragon bishop or, if he feels brave, retreat it to h8, allowing white to play Bxf8, thus sacrificing the exchange. I have seen at least one GM game posted on chess.com where black not only made this sacrifice, but went on to win the game. However, most chess players at the below-2000 level, like myself, rarely have the stomach to make such a daring speculative sacrifice, much less the tactical skill or strategic foresight to do so successfully.
But what about when black's rook is not on f8, what then of the move Bh8 for black, when his "dragon" bishop is attacked? Sure, black values his dragon bishop, and would like to hold onto it, but the problem with this move, as I see it, is that it leaves the black king completely immobilized and thus vulnerable to checkmate, should he leave his back rank open to an attack from white's queen or rooks, or by moves like Ne7 -- a tactic which I personally experienced and lost immediately to in a recent game -- or Bc4 (assuming black has no defense against such moves).
I have personally made this move (Bh6) in two games over the past few weeks, and have lost both of them. Though I made plenty of other errors in these games, It seems that this move had something to do with my loses. I have attempted to post both of these games (emphasis on attempted, this being my first effort at ever posting a game here), and would like feedback, specifically related to the move Bh6 by black. Was this move losing in and of itself in both cases, or were there extenuating circumstances?
I also realize that my games are not the best examples for determining whether Bh6 is a good move for black or not in the Sicilian Dragon and King's Indian Defenses, so if anyone has access to a large database of master games, and cares to take the time to search it for this move, please share and post the games you find, whether black or white won in the end.
P.S. Somehow the text for the body of this post ended up center-justified, though I wanted it left-justified. I have no idea how to fix it, as there are no tools for text justification displayed in the toolbar of my text editor, contrary to what the help menu says. If someone could help me fix this, I'd really appreciate it.