No absolutely not, this keeps your position strong:
Was the pawn loss inevitable?

After 12.Ne5, I don't really see a good way to save your pawn anymore. 12...Qd6 13.Bf4 looks scary. I thought perhaps 12...a6 13.Qa4 0-0 14.Nxc6 Qd7 might work, but it fails to 15.Nxe7 _check_.
Wait... how about 12...a6 13.Qa4 Bd6. If then 14.Nxc6, Qd7 does work. No, it doesn't, White has 15.Re1+ forcing Kf8, and you're probably worse.
Is 11...Rb8 forced? I think so. 11...Qd7 12.Qxb7 Rb8 13.Qa6 Rb6 14.Qa4 leads to nothing and costs a pawn.
So, one more move back. Is there an alternative for 10...Be7? Well, why not 10...a6? I think that works, although it's slow, and White will have a lead in development.
I don't think this is so much a consequence of your exchanges, but of keeping your king in the center for too long. The pin by the queen, the Re1 checks, they're consequences of that.
I don't know this opening, but your ...Bf5 feels a little 'loose'. Perhaps 7...Nxc3 8.bxc3 Bd6 followed by castling would have been better. Get the king off that open file!

No absolutely not, this keeps your position strong:
after a6 he is alright but I think white can still get a slight positional edge by 1.nxc6 and after axb5 2.nxd1. if bxd1 3. r-e1 gives a solid edge. And if 3...rxd1 then 4.r-b1, c3 5. a4 with initiative.

Actually ... best was likely to sacrifice that pawn anyways! :) 14. ... Rb6! and you have a fantastic position!
edit - a6 also works (Nxc6, axb5, Nxd8, Kxd8!) ... your pawn structure is actually quite useful in comparison to whites... there would be a slight advantage to black although very drawish... Rb6 keeps the game fun.

Actually ... best was likely to sacrifice that pawn anyways! :) 14. ... Rb6! and you have a fantastic position!
edit - a6 also works (Nxc6, axb5, Nxd8, Kxd8!) ... your pawn structure is actually quite useful in comparison to whites... there would be a slight advantage to black although very drawish... Rb6 keeps the game fun.
then after r-b8 and b-f4 and a4 seems a pretty similar situation as I described although the game is pretty balanced. whether or not that a pawn was turned into another lost pawn or a pain in whites butt would be the question.

Actually ... best was likely to sacrifice that pawn anyways! :) 14. ... Rb6! and you have a fantastic position!
edit - a6 also works (Nxc6, axb5, Nxd8, Kxd8!) ... your pawn structure is actually quite useful in comparison to whites... there would be a slight advantage to black although very drawish... Rb6 keeps the game fun.
nm mixed up b6 with b3

In my opinion ...
3. ... Nxe4 is better (the white knight is not a threat: who won’t trade a pawn for a knight?)
5. … d5 loses a tempo (instead, develop a piece)
6. … Nxc3 is better (doubles White’s pawns on the c-file and isolates the a-pawn)
7. … Bg5 is better (it denies the capture of Black’s e4 knight by White’s bishop or knight, as long as the pin remains on the white queen)
12. … 0-0 was better than a6, as White’s doubled c-pawns negate his single pawn advantage, if Black pressures this weakness.
My analysis of 12. … a6 is as follows:
the threat of 13. Nxf7 is no threat at all: the black king takes, moves his rook over and then slides his king into the castled position at g8 on the subsequent two moves.
Therefore, 13. Nxc6 axb5 14. Nxd8 Rxd8 15. Re1 rd7 16. Ba3 Kd8 17. Bb4 Bf6

3. ... Nxe4 is better (the white knight is not a threat: who won’t trade a pawn for a knight?)
7. … Bg5 is better (it denies the capture of Black’s e4 knight by White’s bishop or knight, as long as the pin remains on the white queen)
In Petroff 3. .. Ne4 is considered bad in view of
7. .. Bg5 will cost you a good bishop, as your pawns will likely end up on white squares

Yes 3...Nxe4 is not very good, there is some trappy lines where black gets some sort of play for pawn, but not very convincing. 5...d5 is main-line these days, although in past 5...Nf6 was played too. 6...Nxc3 is playable option, but 6...Bb4 maybe the better.

Here's a shocking surprise: you could have won a bishop with a pawn sacrifice!
Actually I think White can take on b7: 11...a6 12.Qxb7 Na5 13.Qb2 Nc4 (otherwise White just moves the Bc1) 14.Qb3 Qd7 (14...Nb6 15.Ba3 or 14...Rb8 15.Qa4+ Kf8, and now, instead of taking on a6, moves like Re1or Ne5 come to mind, when the awkward position of Black's king compensates for the similarly awkward position of White's queen) 15.a4 Rb8 16.Qa2, when I'm unsure how much compensation Black really has.
In the game I seemed to be theoretically sound, until I emerged pawn down out of the opening.
My question: was my pawn loss on move 14 inevitable after I forced an exchange on move 8? Is that exchange a taboo?