Which is also my favorite.

Which is what I also would've done.

I only play the move played.

I would've continued with d2-d4 at once. But, I'm used to the Sicilian Nadjorf.

I find this move most peculiar. Is it theory? (I looked it up: seems it is.)

I feel that didn't have to be defended, and thus, would've played 4.Bc4 since this blocks the center pawn.

I have seen it before. If 4.Bc4 then I can take on e4 as after 5. Qa4+ the Bc4 blocks the attack on e4.

But, he'll get out in a similar way to how I'd do it.

I would've answered with 5.Be2, but I think that this was better.

I disagree about exchanging. As this is what I would've played. I would exchange in the Spanish Opening which is similar, but I don't see a good reason to exchange. And, it's B for N which GM Susan Polgar said is generally not a good idea.

This should be OK for Black, shouldn't it?

Yes

Why didn't he just recapture right away?

Because the knight is pinned by the bishop on h5.

The downside of this option should soon become apparent...  

I would've recaptured with 8...Nxe5.

This was on the cards, anyhow, to break up Black's Q-side.

As the Q would be directly pinned which would allow him to take.

White's game looks a bit tangled, here, but if he unravel the knots quickly enough, he should have a little bit of an edge.