Three Knights Opening. Considered 'leisurely' but can still lead to all sorts of excitement.

(?!) You have some good opening instincts by now (develop, and keep everything defended) but these bishop moves blocking the central pawns should generally be avoided. Preferable was 3... Bb4 (the e-pawn is defended well enough) or 3... Nf6.

Try not to make moves that block your own pieces in unless absolutely necessary. here ...Bd6 is not necessary because your e-pawn is already protected by the knight on c6

This ought to have waited a move or two, I think, upon something like Bb5, Bc4, d3 or d4.

White's move is terrible here. He is moving the same piece twice and slowing down, not speeding up his attack. The simple developing 4.Bc4 looks fine.

4...Nf6 with the same idea would have been OK too.

archduke's move is better

I agree with Joe: the exchange does little to advance White's game. Better might have been something like 5.Bc4, preparing, after 5...Nxd5 to recapture with the bishop.

I agree too, 5.Bc4 is better, and even 5.Ne3 is better looking to reroute the knight to better attacking squares

Good move, leaving the already developed pieces in situ. In this situation, 5...Bxe7 would also have been OK, freeing the d-pawn to advance to d6, or possibly even d5.

I'd prefer 5...Bxe7 admitting my mistake (Bd6) and freeing the d-pawn for rapid advance as needed. The increase in mobility is worth the loss of tempo,

Technically, yes; the bishop can capture the pawn or the knight. In practice we don't bother to give this a special name, as both are defended.

One tends to think of 'forks' as a double-attack. Protected as they are the pawns and the knight aren't really under attack. White could (and does) exchange bishop for knight. Both sides get something from the deal: White pokes a bit of a dent in Black's pawn structure; Black gets the bishop pair. If the board opens up at all, those bishops might well come into their own.

Your move here is called "putting the question to the bishop" exchange or retreat. This is standard, but you had better more aggressive moves here to consider 6...Nd4 or 6..f5. Did you consider either of these?

This exchange is not essential, but not bad, either. See my previous comment.

I would vote no on the exchange. I would not want to hand my opponent the Bishop pair this easily.

If we are going to rate a doubled pawn as 1.5 rather than 2 pawns, perhaps we can rate the bishop pair as 1.25 bishops apiece. I agree that bishop pairs can be overrated, but, given the right conditions, they can really pack a punch.

dxN could have been better in sight of allowing the lsb space to move, countering the effects of the self-shrinking Bd6. With the text, it can move to b7, but then black has to move c5 in order to give it some scope.

damalfi's idea is best. dxc6 and more scope for the Bishops. When analyzing the pawn structure the counting ideas expressed here are mistaken. After dxc6 this is my assessment: Black would have no pawn structure weaknesses and has play on the half-open d-file. White has no pawn structure weaknesses either and both can try for pawn breaks at several points. White can play to enforce d4, aiming for an exchange Ruy Lopez type position.

OK, but I might have been more inclined to play the solid 8.d3 instead, or even castled. White should beware of allowing the board to open up too much, freeing Black's bishops.

...discovering the Q's attack on the e4 pawn!

That will happen as you gain more experience. Both players try to avoid giving away material, so any exchanges offered will tend to be even ones. (Certainly there are situations where it's possible to sacrifice material for other advantages, but at a low level, material loss happens more often by error.)

This move is a serious error. The obvious 9.Qxd4 was quite safe, attacking as it would have done, the g7-square. But best would have been 9.0-0! ... to invite the blunder 9...Qxe4?? to which 10.Re1! would have won the major exchange (queen for rook). After 10.0-0, 11.Qxd4 would have yielded a roughly level game.

Joe's right: this is a fine move - if anything, even stronger than the immediate 9...Qxe4(!?), which also would have won a pawn.

Qxe4+ looks strong because White has to defend the check, and the next move he will lose the g2 pawn to the Q. Nevertheless, if white defends the check with the N (Ne2), black will go into the wilderness: Qxg2, Rg1 (defended by the Ne2). After the Q moves, Rxg7 "returns the favour" to black. True is that the BQ would have grabbed another pawn, h2... but white would have threatened Qg4, supporting his R and setting Rg8+, RxR, QxR+, with a suddenly tactical game. Say: 9... Qxe4+, 10. Ne2, Qxg2!?, 11. Rg1, Qxh2, 12. Rxg7. Black has to play carefully to keep his pawn, e.g. ...Qh1+, 13. Rg1, Qh4.

Had both side been castled, this would have refuted ...c5. But as the kings are still in the centre, Black will pick up the pawn, and force White's knight back to e3 - unless White is willing - a pawn down - to exchange queens.