Black has slightly less space, so trading pieces of equal activity is not a bad deal. After the trades on c3 Black achieves a satisfactory position by playing c5 striking against White's center (probably leading to a position where White has hanging pawns on d4-c4, should be fine for Black), or maybe more promising, by solving the problem of the bad bishop on c8 with b6 and Bb7. White cannot occupy the center right away with e4, because of 11... b6 12. Bd3 Bb7 13. e4 f5! when Black is already better. In both cases White's center has no cramping effect on Black's position because two pairs of minor pieces have been exchanged.
As for 9... Ng5 it's just a bad move. After 10. Ne5 threatening h4 Black is in big trouble.
It's nice to see Stockfish and I are in agreement with Ne4 in the following position. However, why does Stockfish want black to trade off the N? My thinking was that the DSB is pinning the knight and keeping white's DSB occupied. So, I went for Ng5 to break up the kingside pawn structure. Besides opening up the file for the h1 rook are there any other reasons why Ng5 is not good? Would it have been good if say white castled short first and then black played Ng5? What would you have done as black after Ne4? This game is over.