What color do you think will win more often using non-amateur strength?
White will win every single game from this position, "hands down".
What do you think of this position?
Position is better for White, but with "scientific" defensive moves Black might have equal chances (but that requires detailed analysis to be proven for sure).
Let me clarify my answers a bit by analyzing the game after 7. ...Nxd4:
You get 7 units for 6 units ( rook + 2 pawns for knight + bishop ) and mathematically speaking, you are better.
White loses castling, but in the end manages to get his king to safety (the point of castling is to get king into safety, not to just play it) when we add up pros and cons the result is zero. Although he lost castling right his king is safe.
You have doubled and isolated his pawns, so you have better pawn structure.
Your position is not weakened, and you have finished your development.
These are all good things for Black, but here is the "killer":
You have lost 7 tempos (you played b6 + Bb7 and those are 2 lost tempos since you have lost the bishop, 7. ...Nxd4, 8. ...Nxc2 9. ...Nxa1, 10. ...Nd7) to achieve all this, while he lost 4 (8.Qxb7, 9.Kd1, 10.Qc6+, 11.Kc1).
The result is that you lost your "opening investment" while his position has transformed and this transformation has only improved his overall position.
Your position at the start aimed to pressurize White from afar and perform a breakout in the center or on the wing (probably with c5) but now all of your developed units are lost resulting in holes on the light squares which only queen and knight can cover.
In return you have got 2 pawns and a rook, but you need open lines for that rook, and you need to use the extra pawns to gain space, in order for compensation to be real.
White, on the other hand, has "castled" long, his doubled pawns are weak but they open lines for an attack (they do not teach you this in "school") and his opening idea-to physically pressure you by pushing his center and by pinning your pieces, has transformed into physical pressure on the light squares with the promise of the slight, but probably lasting and growing initiative.
Although Black has more pawns in the center he can't grab space with them so they are useless, they hinder his bishop making him bad, and White's e3 pawn controls d4 so there is no way we can say that Black has an advantage in space, rather it is a nearly equal distribution of strength for both.
You have developed more units in the beginning (by sacrificing space, hoping to attack from afar) , but now you have 3 developed pieces (castled rook, bishop and a knight), but White has developed his queen, bishop and a knight so you are both equal in that view.
This means that you have deliberately lost your development advantage.
Mathematically speaking, you have an "extra pawn", so we need to see the compensation/positional/tactical factors, to properly assess this position:
White's opening strategy was to physically occupy the center (gaining space advantage) and to pressurize early (he played early with queen trying to do some sort of a pin), sacrificing his development.
Black's opening strategy was to develop fast and counter when feels comfortable.
The transformation of the position left you with rook + 2 pawns vs 2 minor pieces.
In those positions the side with minor pieces strives to leave the queens on board, exchange rooks, and go for mating attack, while the side with the rook strives to remove the queens and open the position, reaching an early endgame (you have 2 extra pawns).
In the above position, white has strong pressure, that can turn into mating attack (he already has h3 so he needs g4 and there we go!), while there is no way for you to further simplify and open lines.You can not even grab some space with the extra pawns you have.
The only way I see for you here (and Houdini confirmed it) is to sacrifice a pawn to open position and try to simplify things but this would mean that mathematically the material would be equal (pawn + rook vs 2 minor pieces).
Furthermore, it would bring you new weaknesses, without you losing the existing ones, and still I do not see what/how will you attack/create counterplay.
Even though your "math" was correct, you have a flaw: you miss-evaluate the resulting position after the exchange.You need to find the book on that, because this is a serious flaw that will cost you a lot in the future (I was there, believe me). At the moment I can't remember any book in English, but look it up. Just in case you fail to find books, I will try to give you a piece of advice for future games:
When exchanging pieces, always evaluate the resulting position that will arise after it.
If you only had known this earlier, you would have stayed away from 7. ...Nd4, because you would see all the cons and pros I have listed above.
Hopefully this will help you, if you have further questions, feel free to ask.
Good luck in your future games!