Nice game.
I know from playing the Danish as white that you generally want the pieces to stay on the board and avoid exchanges but 7.Bxf7+ followed by Qd5+ picking up the knight on e4 looks good to me. 7.Nf3 has to be a mistake if you follow 7...Bb4+ with 8.Nbd2 since this allows further simplifications. 8.Kf1 is always an option in the Danish since it allows a rook to swing over to e1 (and avoids exchanges!) where it can attack along the (open) e file if blacks king is in the centre but unfortunatley that is not the case here.
After move 10 black is comfortable but the power of Whites bishops can still lead to mate (as in the game). Giving back a pawn is usually a good idea for Black against the danish so ...d5 is good, better than ...Nc6.
12.a3? White has no time for that, why attack a lonely bishop when the kings head is on the line!
Ok, so White has got away with blacks greed (lack of ...d5) but needs to pull off this attack precisely. 13.Qg4 looks like a 'blitz' move, but far better is 13.Qh5! attacking the weakend f7 (because ...Re8) and carries the threat of Ng5 with serious threats against f7 and h7. White winning after that.
Nevertheless, Black showed his greed and nice mate, a lucky one at that, for white. To Decoy321 i would suggest looking through some Danish games played by masters but also ones against weak black players because you, as white, need to know how to take advantage of blacks errors. The advantage can swing from side to side in the Danish and attacking precisely is paramount for white. For example, if you read the art of attack you would have spotted Qh5 (+-) attacking 2 weak points rather than the illusionary threat of Qg4 (=).
Let us know how the Danish goes for you Decoy321 in your next games. For me, the danish has brought me wins against all strengths of players and hope the same for you.
Recently I read one of the articles titled "Greed doesn't pay" and decided to play the opening a few times myself, hoping someone would accept this gambit. What's funny to know is that Lasker even lost a match in 12 moves when accepting this gambit against Bird. http://www.chessgames.com/perl/chessgame?gid=1028156
Here's the game. Look at black, greed in his 14th move by taking my bisshop kills him :)