Petrov gambit? Speculative by black, but I won!
Sort:
Meh, Nc3 looks fine to me. d3 is certainly good too... it's 6.a3 that loses the pawn for no reason.
Be2 is not a bad spot for the bishop. The Qe7 Be2 sequence is pretty mundane IMO :p
Qe3 doesn't gain you the center, it challenges the e-file. You're not threatening Qxe3 (only black can make that decision). I mention this because assuming a capture or re-capture is a common mental mistake.
Qxe2 blunders the rook, and you captures it so that's good.
Bh3 blunders the king, and you mate him so that's good.