rooked!
Be2 is not really developed.
Well, in the sense that it gets me one move closer to castling...

If the Black played Qe6 +, he probably wanted to trade Queens. You should avoid it and instead show that his Queen is misplaced there, at least that's what I'd do.
Classy finish, those rooks were the stars of the show. During the attack against the king, it really seemed like a c5 break would have helped very much. At move 16 in particular, your plan was reasonable, but c5 just makes Blacks pieces look ridiculous. At move 19 as well, c5 looks pretty devastating.
And as the other guys mentioned, swapping queens so early is probably what Black wanted and it doesn't help you at all. 4. Be2 is the move and maybe your minor pieces can give the perhaps overexposed Black queen some grief in a little bit.
Right you are about c5. I think I had an idea that I might bring my rook to a8 (although hadn't thought much beyond that), but c5 would have achieved my goal of rescuing the bishop and exposing the king in a much more direct way.
The comments about exchanging queens are interesting to me, as I really thought I came out slightly ahead by move 5, but I will bow to the superior wisdom of the group. As a general rule of thumb, not doing what your opponent expects makes sense.