I went through all the variations and it looks like black has equalized in all lines with best play. 10...Qe7 is much better than the previous 10...Qc5. White's only hope lies in the endgame with two rooks versus the queen. Note how the following line is similar to the one I gave after 10...Qc5 except the queen is better placed.
1.e4 e5 2.Nf3 Nc6 3.Bc4 Nf6 4.Ng5 Bc5 5.Nxf7 Bxf2+ 6.Kf1 Qe7 7.Nxh8 d5 8.exd5 Nd4 9.d6 Qxd6 10.Nf7 Qe7 11.d3 Bg4 12.Qd2 Bh4 13.Nc3 b5 14.Nxe5 0-0-0 15.Nxg4 Rf8 16.Ne5 Ne4+ 17.Qf4 Rxf4 18.Bxf4 Nd2+ 19.Bxd2 Qxe5 20.g3 bxc4 21.Kg2 Bf6 22.dxc4
White may be able to squeeze something out of the endgame, but it will be difficult against black's activity. Thiss is better than 15.Nf3 because that leads to a favorable endgame for black.
1.e4 e5 2.Nf3 Nc6 3.Bc4 Nf6 4.Ng5 Bc5 5.Nxf7 Bxf2+ 6.Kf1 Qe7 7.Nxh8 d5 8.exd5 Nd4 9.d6 Qxd6 10.Nf7 Qe7 11.d3 Bg4 12.Qd2 Bh4 13.Nc3 b5 14.Nxe5 0-0-0 15.Nf3 Nxf3 16.Qe3 Re8 17.Qxe7 Rxe7 18.Bf4 bxc4 19.h3 Bh5 20.gxf3 cxd3 21.cxd3 g5 22.Bd2 Bxf3 23.Rh2 Bg3 24.Bxg5 Bxh2 25.Bxf6 Re3 26.d4 Bb7 27.Re1 Rf3+ 28.Kg2 Rg3+ 29.Kf1 Rxh3 30.Re8+ Kd7 31.Rb8 Bh1 32.Rh8 Ke6 33.Be5 Bxe5 34.Re8+ Kd7 35.Rxe5 Bb7 36.Na4 Bc6 37.Ra5 Rh4
White is fighting for a draw, black has a modest advantage in the endgame.
The problem is that the resulting position will leave two rook pawns, opposing b and c pawns, Rooks, and the minor pieces on the board. Even though white is fighting for a draw, it should still end as a draw based on the structure of the position. Black's minor piece advantage is his only hope for a win.
I've only looked at 10.Qe7!?, Jerz' suggestion. Variations C, E and F lose for white, B and D are perpetuals. White may try for an advantage in A (A2, for instance, is not clear, but not very promising either) and H, which may be white's best try.
1. e4 e5 2. Nf3 Nc6 3. Bc4 Nf6 4. Ng5 Bc5 5. Nxf7 Bxf2+ 6.
Kf1 Qe7 7. Nxh8 d5 8. exd5 Nd4 9. d6 Qxd6 10. Nf7 Qe7 (A)11. d3
(B)11. c3 Bg4 12.Qa4+ Nd7 13. cxd4
(13. Kxf2 Qh4+ 14. Ke3
(14. Kf1 Bh5 15. Nxe5 O-O-O 16. Nxd7 (16. cxd4? Rf8+
17. Nf3 Bxf3 18. Qa3 (18. gxf3 Qh3+ 19. Ke1 Qxf3 20. Rf1
Re8+) 18... Bxg2+ 19. Kxg2 Qf2+ 20. Kh3 Rf3+ With a winning
initiative.)(16. g3 Qf6+ 17. Kg2 Bf3+ 18. Nxf3 Qxf3+
19. Kh3 19... Qh5+)
16... Qf4+ 17. Kg1 Be2 18. h4 Nf3+ 19. Kf2 Nd4+ 20. Kg1)
14... Kf8 15. Qa3+ c5 16. Nd6 Qg5+ 17. Kf2 (17. Ke4 Nf6+ 18. Kd3 Be2#)
17... Qf4+ 18. Kg1 Be2 19. h3 Nf3+ 20. Kf2 Nd4+)
13... Qf6 (13... Bxd4 !?) 14. Nd6+ (14.dxe5? Qf4) 14... cxd6 15. dxe5 Qxe5
(15... Qf4? 16. Bf7+ Qxf7 17. Qxg4) 16.Bf7+ (16. Kxf2 Qf4+) 16... Ke7 17. Kxf2
Rf8 18. Qxg4 Rxf7+ 19. Qf3 Qd4+ 20. Ke2 Rxf3 21. gxf3 Ne5 22. Rf1 Qc4+ 23. d3
(23. Kf2 Nd3+) 23... Qxd3+ with perpetual.)
(C)11. Nxe5? Qxe5 12. c3
(12. d3 Bh4 13. Qd2 Nxc2)
(12. Kxf2 Ne4+ 13. Kg1 Qc5)
12... Bg4 13.cxd4 (13. Qa4+ Kd8) 13... Qf5 14. Be2 Bh4+ 15. Bf3 O-O-O 16. h3
Qd3+ 17. Kg1 (17. Qe2 Re8) 17... Qxd4+ 18. Kh2 Qd6+ 19. Kg1 Re8 20. Bxg4+ Kb8
21. Nc3 Qd4+ 22. Kh2 Qf4+)
(D)11. Ng5 Bg4 12. Nf3 e4 13. h3
(13. Kxf2? O-O-O 14. Re1
(14. h3 exf3 15. hxg4 Nxg4+ 16. Kg3 Ne3 17. dxe3
(17. Qxf3 Nef5+ 18. Kf2 Nxf3 19. gxf3 Qc5+ 20. Kg2 Qxc4)
17... Qg5+)
14... Qd6 15. Re3 Bxf3 16. gxf3 Qxh2+)
(13. Be2? exf3 14. gxf3 Bh4 15. fxg4 O-O-O With a typically strong attack.)
13... exf3 14. hxg4 fxg2+ 15. Kxg2 Qe4+ 16. Kxf2 Nxg4+ 17. Kg1 Nf3+ 18. Kf1 Nfh2+
19. Kg1
(19. Rxh2? Qf4+ 20. Ke1 Qg3+ 21. Ke2 Qxh2+ 22. Kf3 Kd7 23. Kxg4 Rf8 With
a mating attack.)
19... Nf3+)
(E)11. Kxf2? Ng4+ 12. Ke1 Qf6 13. Rf1 Qh4+ 14. g3 Qxh2 15. d3 Qxg3+ 16. Kd2 Qe3+
17. Kc3 Ne2+ 18. Qxe2 (18. Kb3 Qb6+ 19.Ka3 Qa5+ 20. Kb3 Nd4#) 18... Qxe2 19. Rh1
Qf2 With a winning initiative.)
(F)11. b3? Bg4 12. Ba3 Qd7 13. Nxe5 (13. Qc1 Qf5) 13... Qf5 14. Nf3 Nxf3 15. gxf3
Bxf3 16. Qc1 Qh3+ 17. Kxf2 Ng4+)
(G)11. h3!? Bh4 12. c3 (12. Nc3 Nh5) 12... Nf5 13. Qb3
(13. Na3 Ne4 14. d3 Qf6 15. Qf3 Nfg3+ 16. Kg1 Ne2+ 17. Qxe2 Bf2+
18. Kh2 Bg3+)
(13. d4 exd4 14. cxd4 Be6 15. Bxe6 Qxe6 16. Ne5 O-O-O 17. Nf3 Ng3+
18. Kg1 Ne2+ 19. Kf1 Ng3+)
13... Qc5 14. Qb5+ Qxb5 15. Bxb5+ Kxf7 Black has an initiative,
but this may be good for white.)
(A ctd.)11... Bg4 12. Qd2 Bh4 13. Nc3
(13. Qe3 Be2+ 14. Qxe2 (14. Kg1 Ng4) 14... Nxe2 15. Kxe2 b5 16. Bb3 c5
The knight on f7 is in trouble.)
13... b5!! 14. Nxe5!
(14. Bxb5+ Nxb5 15.Nxb5
(15. Ng5 Nd4 16. h3 Qf8 17. Kg1 Qc5 18. Kf1 (18. Qe3 Bxg5)
18... Qf8 A very interesting perpetual.)
15... Qxf7)
(14. Nxb5? Ne4!! 15. dxe4 Qf6+ 16. Kg1 Ne2+ 17. Qxe2 Bxe2 18. Nxc7+
(18. Be3 Bxc4 19. Nxc7+ Kxf7 20. Nxa8 Qc6)
18... Kd7 19. Be3 Bxc4 20. Nxa8 Qxf7 21. g3 Qg6 22. Rd1+ Kc6)
(A ctd.)14... O-O-O! (A1)15. Nf3
(A2)15. Nxg4!? Rf8 16. Ne5
(16. Nf2 Bxf2 17. Be6+
(17. Kxf2 Ne4+ 18. Kg1 Nxd2 19. Bxd2 Qc5)
(17. Qxf2 Ng4 18. Be6+ Nxe6 19. Qf3 Nd4)
17... Nd7 18. Bxd7+ Kxd7 19. Qxf2 Rxf2+ 20. Kxf2 Qf6+ 21. Kg3
(21. Ke1 Nxc2+) 21... Qg6+ 22. Kf2 perpetual. (22. Kh3? h5
23. g3 Qf5+ 24.Kg2 Qf3+ 25. Kg1 h4 26. gxh4 b4))
16... Ne4+ 17. Qf4 Rxf4+ 18. Bxf4 Nd2+ 19.Bxd2 Qxe5 20. g3 bxc4 21. Kg2 Bf6 unclear)
(A3)15. Nxb5? Qxe5 16. Nxd4 Rxd4 17. Kg1 (17.c3 Ne4 18. Qc2 Nxc3 19. bxc3 Qe1#)
17... Ne4 18. Qe3 (18. Qf4 Bf2+ 19. Kf1 Ng3+) 18... Bf2+ 19. Qxf2 Nxf2
20. Kxf2 Rf4+ 21. Bxf4 Qxf4+ 22. Kg1 Qe3+)
(A4)15. Bxb5? Qxe5 16. Qe3 (16. h3 Qg3 17. hxg4 Rf8) 16... Qf5+ 17. Qf4 Qe6 18. Qe3
(18. Bd2 Nh5 19. Bc4 Be2+ 20. Kg1 Qb6 threatening Nf3+. 21. Be6+ an
interesting resource, blocking the 6th rank.
(21. Qe3 Nf3+ (21... Bg5 22. Qf2 Bxd2 23. Nxe2 Nxe2+))
21... Kb8 (21... Nxe6+ 22. Qe3) 22. Qe3 Bg5! 23. Qf2 Bxd2 24. Nxe2 Nxe2+
25. Kf1 Qxe6)
18... Qf7 19. Kg1 (19. Qf4 g5) 19... Nxb5 20. Nxb5 Re8)
(A5)15. Qe3!? Nxc2 16. Qxa7 Qxe5 17. Qa6+ Kd7 18. Qxb5+ (18. Bxb5+ ?? Ke7)
18... Qxb5 19. Bxb5+ c6 20. Ba4 Rf8 21. Bxc2 Nd5+ 22. Kg1 Bf2+ 23. Kf1 Bh4+)
(A1 ctd.)15... Nxf3 16. Qe3
(16. gxf3? Bh3+ 17. Kg1 bxc4 18. Qe3 Ng4 19. fxg4 Rf8 20.Bd2 Qf6 21. Qf4
(21. Qe2 Bf2+) 21... Qb6+ 22. d4 Rxf4)
16... Re8 17. Qxe7
(17.gxf3 Qf8 18. Ne4 Nxe4 19. dxe4 Bxf3 20. Qf4 Qxf4 21. Bxf4 Bxh1 22. Bxb5 Rxe4)
17... Rxe7 18. Bf4 bxc4 19. h3
(19. gxf3 Bxf3 20. Rg1 Ng4 21. dxc4 Rf7 22. Bg3 Bg5 23. Ke1 Re7+ 24. Kf1 Rf7)
19... Bh5 20. gxf3 cxd3 21. cxd3 g5 22. Bd2 Bxf3 23. Rh2
(23. Rg1 g4 24. hxg4 Nxg4 25. Bf4 h5)
23... Bg3 24. Bxg5 (24. Rf2 Bxf2 25. Kxf2 g4) 24... Bxh2 25. Bxf6 Re3 26. d4 Bb7
27. Re1 Rf3+ 28. Kg2 Re3+ 29.Kf2 Rf3+ 30. Kg2