Your opponent took your knight for free on move 5. You need to recapture.
On move ...9 you have a combination, 9...Nxe4 10 dxe4 followed by 10...Bxf2+.
After 10...Nh5 you opponent can win another piece with no compensation to you after 11 d4, when you have two pieces hanging, and no threats.
I agree that 13...g6 is not good. This makes Black's kingside weaker and subject to attack by the White h pawn. A better try might be Bg3 with the idea Rf2+.
I played as black. I know for a fact that I made quite a few mistakes, and I'm pretty sure white made a few as well. I was down in material for most of the game...I sac'ed my knight on move 10 with the intent on converting it to a positional advantage, but I don't feel like I accomplished it. Any help at all would be appreciated.