This is pretty good! It blocks the line of attack for the c4 bishop, ...e5 would weaken the d5 point and the d7 paw. Black's idea in this opening is to play ...d5 and ...e6 is a preparatory move to play that.

Yes, this certainly looks a better move than ...e5 to me.

3...e5? is a bad move. It creates an outpost on d5 for white Knight. It also prevents black from playing d5 later because white has control of this square. However, 3...e6! is a good move as it blocks the white bishop, allow d5 and prevent white from playing Nd5. It is the normal move in sicilian after white plays Bc4.

I think better is 5.d4 or 5.O-O. if 5.d4 cxd4 6.Nxd4 Nf6 7.f3 (or e5!) then Ndb5 followed by Nd6+and white has the advantage. Opening the d-file is very good for white to occupy it with a rook, that is why d4 is better.

The text move implies white relies on the Bf4 - Nb5 plan, "looking" to d6 and c7. This is not a bad idea for white, but black has no problem defending from it unless something more is thrown into the fight.

7.Bb5+ now gives white a big lead in development, with Ne5 and O-O coming next.

This move is useless. It only allows black to continue his development. Much better was 7.Bb5+

7...Be2 is the correct way to respond, as now black can play ...Nf6 and ...O-O, getting the K to safety. After 7...Ne7 and 8...d4, black is probably lost.

You mean Be7. And white must have thought on Bb5+. It is kind of naive, tough.

9.Ne5 wins on the spot. Black has a slight weakness on f7 that is problematic in this position.

9.Nb5? is a bad move. Instead, 9.Ne4! gives white a great advantage. He can then play 10.Neg5 attacking the weak square on f7. It may also allow 10.Nf6+ if black made a mistake or even sacrificing the knight. or playing 10.Bf4 followed by 11.Nd6+ wins.

I did this to pin the queen since it is the only piece protecting the knight fork.

This is good, but if you already have the B on f4 so if black plays a6 you can go to c7 or d6. but now you have to retreat the N to a3... however, Ne5 threats mate in 1 (Bxf7), so black must go Bd5 to defend (or move the queen), and then NxBd5 is possible (if Ne7xNd5, Nc6+ (discovered) wins the black queen). You were lucky your opponent didn't play a6 here, but you cannot rely on such overlooks.

10...Bg7, making room for the K to run, is probably best, though black shouldn't survive much longer in any case.

Now appers the advantage of 9.Ne4 as if black played 9...g6?? 10.Nf6#. Here, better for black was 9...a6 to drive the knight away then 10...g6, 11...Bg7 12...O-O. and black stands better than now.

At least you spotted correctly the plan to reinforce the attack and the knight possibilities with the Bf4. But it was the wrong order. The bishop cannot be easily drown away, the knight could (not anymore).

Actually this isn't the best move. White should play 11.Bxh7+!! if 11...Kd7 12.Qe6#. if 11...Kxh7 12.Nd6+! Kf6? 13.Qe5# if 12...Kg8? 13. Qe6+ Kg7 14. Qf7# if 12...Kg7? 13.Qe5+ Kg8 14.Qe6+ Kg7 15.Qf7# if 12...Qxd6 13.Bxd6 wins easily.

I have a hard time finding long term checkmates when there are many different branches that the sequence can go in. I wouldn't have ever found that move in the time limit I had.

12. Ne4+ was also winning, and quicker. If KxN, Nxf7+ (disc) wins the black queen or the game: try it by yourself. Other king moves from black are not much different.

Nice mate at the end.

I agree. Nice.