In the Dubois-Steinitz match (1862), at what point does the match become irreversibly lost for White? And why does the engine not pick it up early? I am a beginner player (~1400 rating on lichess). I was studying a famous match between Dubois and Steinitz shown below (analysis done by Stockfish on lichess.org). My doubts are:

1) At what move does the game become "bad" for White? Is it the 9. Nxg5 move? Or is it some move further downstream? After Black plays 12...Nd4 was there any possible move that would have saved White?

2) Why doesn't the Stockfish engine pick up the fact that Black is winning until the very last moment when White plays 13.Nc3 ? According to engine analysis, 8...h5is an inaccuracy for Black, 9.Nxg5 is a mistake for White, and 10...hxg3 is a mistake for Black! More surprisingly, till 12...Nd4 the engine considers White slightly better (or rather neutral) at +0.3, and in the White's next move it suddenly says checkmate is unavoidable and suggests  13.Ne6 to be better for White. But even with 13.Ne6, black can still play 13...Nf3+ and so on, and win the game. 

[fen ""]
[Site "https://lichess.org/OtJ2x5RY"]
[Date "London 1862"]
[White "Serafino Dubois"]
[Black "Wilhelm Steinitz"]
[Result "0-1"]
[ECO "C50"]
[Opening "Italian Game: Giuoco Pianissimo, Normal"]
[Termination "Normal"]
[Annotator "lichess.org"]

1. e4 e5 2. Nf3 Nc6 3. Bc4 Nf6 4. d3 Bc5 { C50 Italian Game: Giuoco Pianissimo, Normal } 5. O-O d6 6. Bg5 h6 7. Bh4 g5 8. Bg3 h5?! { (-0.08 → 0.50) Inaccuracy. Best move was Na5. } (8... Na5 9. Bb3 Nxb3 10. axb3 a6 11. c3 O-O 12. Nbd2 Ba7 13. Nc4 Be6 14. Ne3 Kg7 15. b4) 9. Nxg5? { (0.50 → -1.08) Mistake. Best move was h4. } (9. h4 Na5 10. Nbd2 Bg4 11. Bb3 Nxb3 12. axb3 Bxf3 13. Nxf3 g4 14. Ng5 Qe7 15. c3 O-O-O) 9... h4 10. Nxf7 hxg3? { (-0.83 → 0.38) Mistake. Best move was Qe7. } (10... Qe7) 11. Nxd8 Bg4 12. Qd2 Nd4 13. Nc3?? { (0.31 → Mate in 7) Checkmate is now unavoidable. Best move was Ne6. } (13. Ne6 Ne2+ 14. Qxe2 Bxe2 15. Nxc7+ Kd7 16. Nxa8 Bxf1 17. Bb5+ Ke7 18. Nc3 Bxd3 19. Bxd3 gxf2+) 13... Nf3+ 14. gxf3 Bxf3 15. hxg3 Rh1# { Black wins by checkmate. } 0-1
Playing the game on lichess.org, I only need to wait for a few dozens of seconds to reach the following analysis for White ninth move: 

9. Nxg5 is an interesting counter-attack move that ultimately fails to bring an advantage but white is not lost. After 12. ... Nd4 13. h3! Ne2+ 14. Qxe2 Bxe2 15. Ne6 White survives.

Since 13. h3! saves White, 13. Nc3?? is indeed a fatal blunder. 13. Ne6?? also yields a mate in 7 for black.