'stead of your #4 move you can go for symmetrical
Four Knights Opening defense idea
I don't like symmetrical variations as black past the third move because it puts black down a tempo.
Turns out this is the best move after Bc4. It's under Four Knights' Game: Italian Variation. Now that I've found out it's the 2nd. most popular response to 4. Bc4 (a few thousand games or so have it), I feel foolish for thinking it was an new idea (I should know by now that there aren't any of those left). Either way, Black wins 60% of the time with 4. Bc4 ... Nxe4, which means, of course, that I'm not going to play Bc4 again as white.

Correct, this variation is the reason why 4.Bc4 is bad (though you also have to be prepared for 4.Bc4 Nxe4 5.Bxf7+).
White's main moves in the Four Knights are 4.Bb5, 4.d4 and 4.g3.

Please point out everything you see wrong with this- I want to be able to a four knights game as black. It's frustrating to play Petrof's only to see white play 3. Nc3. Here's what I have for the Four Knights Game so far:
How about 3.... d5! I think you've got a great chance to fight for the iniative there.

after ...4.nxe5 what about 5.Bxf7+ Kxf7 6.Nxe4 looks okay for white to me...
Not really. I mean it's playable, but the main line is better. Black's king is relatively safe, and he will castle manually. Black also has the bishop pair while white doesn't and white has no real compensation.
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Please point out everything you see wrong with this- I want to be able to a four knights game as black. It's frustrating to play Petrof's only to see white play 3. Nc3. Here's what I have for the Four Knights Game so far: