When it was available, I was looking at your knight taking on G2. Assuming king takes, then you snag the H pawn with check and skewer the king and rook. The net result is you gain a rook and 2 pawns for your bishop and knight, but your attack has probably fizzled. I stopped looking at that line at that point. It looks like you now have converted your attack on the king into some potentially very strong passed king side pawns of your own to exploit. Good catch at the end though. Blunders are always accepted graciously.
i think the main line against the vienna is Nf6 after Nc3, then if they play the f4 you reply d5. then they will fxe5, and you can reply Nxe4. then white will play Nf3 or Qf3 but black has equalized according to the engine at this point and should be looking to force an endgame with a correctly timed Qh4+. i think if black misses the opportunities to force an endgame then the positions can be tricky to navigate.
in this mirror setup i don't think they should be playing f4 so early. i think Bc4 or Nf3 is better. Bc4 can be fun if black continues to mirror. there is Bc4, Bc5, Qg4, Qf3, Nd5, Qxf2+, Kd1 and this is good for white.
opponents opening play really doesn't make any sense. if you are going to play Bb5 in order to trade the bishop with the knight you need to take the pawn if its offered. Bb5 in vienna is multipurpose so maybe opponent played it for another reason and didn't notice the pawn was hanging.
Bb5 in vienna is for:
a) potential threat to trade with knight and win the e5 pawn. b) immobilising the knight so you don't have to deal with Bg4, Bxf3, Qxf3, Nd4 nonsense c) damaging the structure and playing against the double pawns. so you will often see Bxc6 followed by b3. especially if the pawn is still on d2 because white doesn't have to worry about weakening the c3 knight or creating this hole on c3.
there shouldn't be any risk from taking the pawn and the king getting stuck in the center because white can play b3 and easily castle queenside in this position.
Why are you speeding through you analysis so much? I don't know what part of your game you are focusing on at the moment, but you didn't look at anything. In the game you said you always forget what to do, why not figure out that in the analysis.
The answer is exf4. Your choices are actually quite narrow, moves like 3...Bc5 or 3...d6 allow White to get the position they want, so there are no interesting options besides taking on f4 for Black. However that requires study, like quite broad opening study, so if you really want to avoid that and not get bad positions, another somewhat interesting approach is taking on f4 after Bb5, to then break the centre with d5. For example 1. e4 e5 2. Nc3 Nc6 3. f4 d6 4. Nf3 Nf6 5. Bb5 exf4 6. d4 a6 7. Be2 d5 In case of 7.Bxc6+ the position seems solid after Be7-O-O.
After the horrible opening you handled the middlegame very well starting from Nh5 and then f5. If you aren't interesting in openings, this middlegame analysis would be also great. Like the briefly mentioned c5 with the only purpuose being to stop d4. Or understanding the difference between Rf6-Rg6 and g5. Or looking at that Nxg2 line earlier (which I don't think is a good practical choice, I really like Nxe2-f5). Also I think Rg3 blundered Qxg3
Overall your opening was very bad, but you completely demolished your opponent in the middlegame. I don't know if you knew in those positions where black has that bad queenside structure the play fully shifts to kingside, where Black has to find something and has to do it fast to not get completly locked out. Your play was very fast and very logical.
I don't know how are your other lines against 1.e4 holding up and I don't know if you are willing to spend time on openings, but in case you don't, an interesting option might be the Lion Pirc/Philidor , where you play e5 d6 Nf6 Be7 O-O and generally Nbd7-c6, so it's impossible to get tricked in the opening, since the position is so healthy and solid, while keeping the position similar to typical e4 e5.