What about 10...d5 preventing the immediate Bf7+ and threatening ...Bg4 attacking their queen?
Votechess Problem Line
Oh, you beat me to it, Conquistador. As you say here and I do in the game itself, 13...Qf4+ seems like a pretty straightforward draw. We can hardly hope for more in this line.

10...d5 looks very strong for black. 11.Nxd5 Ne4 leads to a rapid mate, and 11.Bxd5 Bg4 and they lose their queen. I suspect that 10.Nc3? was just a mistake by them.

Well 10...d5 seems pretty strong. The natural continuation 10...d5 11.Bxd5 Bg4! turns the tables on white. Good find!
11.Nxd5 is met very strongly by 11...Ne4!
11.Bb5+ c6 12.hxg3 Qxg3+ 13.Kf1 is unclear, although black looks like can keep the draw.
11.hxg3 Qxg3+ 12.Kf1 Bh3+ 13.Rxh3 Qxh3 14.Kf2 dxc4 and black has the advantage.

Offhand I don't see a way for them to save the game after 10.Nc3? d5!
11.hxg3 Qxg3+ 12.Kf1 Bh3+ 13.Rxh3 Qxh3+ 14.Kg1 dxc4 gets us a rook and bishop, and we could follow up by castling queenside and bring our rook into the attack.

After 11.Bb5+ c6 12.hxg3 Qxg3+ 13.Kf1 Bh3+ 14.Rxh3 with the same plan I think black is winning as well.
Here is the line we had in one of our votechess games. This thread is for the sole purpose of finding improvements in the event we play this line again so that we do not face the same difficulties.
1.e4 e5 2.Nf3 Nc6 3.Bc4 Nf6 4.Ng5 Bc5 5.Nxf7 Bxf2+ 6.Kxf2 Nxe4+ 7.Kg1 Qh4 8.g3 Nxg3 9.Nxh8 Nd4 10.Nc3 d6 11.Bf7+ Kd8 12.hxg3 Qxg3+ 13.Kf1 and now
We played 13...Bh3+ 14.Rxh3 Qxh3+ 15.Kf2 Qf5+ 16.Kg3 which leads to the problem we now face fighting for a draw.
I suggest two improvements.
First I propose 15...Qh2+ to prevent the queen defense although this can turn out badly for black if white can escape with 16.Ke3.
Second I propose 13...Qf4+. By keeping the bishop at home, the queen cannot interpose the checks so perpetual check will be forced.