This is the dragondorf variation (if I'm ignoring the fact that your opponent somewhat inaccurately delayed Nf6,) a mix between the najdorf and the sicilian dragon, combining a6 and g6. It's not very common. In terms of the opening, based on the fairly standard set up you choose (going after the fienchettoed bishop) it is more usual to insert 0-0-0 and f3 before committing to Bh6. The pawn set up with f3 generally works better against dragon like formations than pushing f4, which you see more in other variations of the sicilian. The pawn on f3 supports both the center pawn on e4 (which you had some trouble with in this game) as well as the kingside pawn thrust g4 (it also keeps the black knight out of that square which can be annoying for white in some cases.) So the kinside attack structure you want is f3, g4, h4, and then procede based on the individual situation. The reason I previously said to avoid committing to Bh6 so early, is that sometimes the pawn attack takes priority (this is often the case in the dragondorf.) There are occasionally other ways to attack the crucial darksquared bishop too. I saw in one dragondorf game, after white had f3, g4, h4, black then pushed h5, white responded with g5, temporarily locking the kingside and gaining space. White then moved his knight from d4 to b3, and placed his dark bishop on d4 where it was traded with black's. White won that game quite convincingly. Other times the pawn storm can take place without trading of dark squared bishops. Another note on your pawn storm: maybe I would have played h4 before pushing g5 to retain the flexability of having both g5 and h5 at your disposal.
On to specific inaccuracies: b4 is poor. Playing on the wrong side of the board, and not even defensively. I think a well timed e5 by black could have exposed the disadvantages of this move, cracking open the center in a favorable way for black and pressuring weakness (the a1 to h8 diagonal, the queen on the e file.) Black's kight, queen and rook spring to life. Another thing, you're right that Nf8 is horrible and drops a knight but you allowed him to save it with accurate play. By choosing to trade bishops first you allowed him to play Nxb7 saving the knight as opposed to what he actually did, Qxb7.
As for a positional advantage, the position is very complex and both players have weak pawns/files that needed to be pressured. Your c and f files were a little weak because of the backward pawns. His e pawn was isolated and very weak (which is why he probably thought he had to resort to moves like Nf8) and so was his h file (because it had been cracked open by your kingside attack,) and I would have like to have seen you use your rooks to pressure those locations. I would have also liked to have seen your knights become more dynamic (there were ideal supported posts on c5, e5 and f6.) Obviously your ability to do these things was cut short by your opponent's blunder, so you're not entirely to blame. Despite the fact that more initiative could have been taken on both your parts, you were positionally superior at some points due to the aforementioned weaknesses your opponent had.
You played this game well, and I hope this helps,
Please ask me if you have any more specific questions,
Andrew
I really need some advice on a few things. How did I handle the pawn advance against blacks king? Could I have done a better job with any of the opening moves? Was I positionally superior at any point during the game and why? What Scicilian Variation is this? Najdorf? Yugoslav? Please explain and point out any missed tactics or poor moves that weakened my position. Thank you in advance!