Normal 1.c4 openings have a way of beginning with this move...

Strategically compromising: relinquishing control of central squares, placing the pawn on a vulnerable square, and wasting time better spent doing something else ( 22.Nc3, say).

Odds giving style?

Pretty odd and I'd say bad move by black.

So this was black's plan after bxc.

Chess is not the right battleground for this kind of kamikaze attack. I miss it too...  

Looks like bughouse where you give your partner the pawns he needs to checkmate  

You reckon??

White has just dropped two pawns from his initial position. Black is... more developed, maybe?  

well, this was all right indeed. The weak c6 is black's best target.

Given the absence of center's white control, g6 and Bg7 looked really strong...

Developement is an advantage... if you have a plan behind it. Black doesn't, and this was partly because of your early random moves. That worked for you this time, but it won't with better players, be aware... they just will play solidly, and crush you. Said by experience...  

Better was 8...Nb6.

Good move.

10.Bxd7ch might have been interesting: 10...Kxd7 11.e4 Nf6 12.Qxf7ch.

Black must take some action against white's pieces. The Q forced to defend the B, and the K forced to defend the N (not defended by the Q because she's dued to the B defence...). As you correctly point, Nxe3 does a lot to solve black's problems, as N5-b6, or N5-f6. Black just "ignores" his weaknesses and wants to castle asap. I will just give white another target, since black has no piece mobility on the kingside, where his K will be. - Nc5 can be playable? It requires some safety calculation, but it does seem so.