Leaving yourself in a pin = bad

And my opponent punished me for it.

5. ... c5. What's so great about making the queen move again? This move hinders the mobility of your f8 bishop and concededs the d5 square to white.

"7. ... d5. Looking to open up the center even more" Why are you trying to open the center? This move seems to just give away a pawn.

 "I'm down a pawn... but chances are still fairly even." Why would chances be even if you're down a pawn. I would say that white has a good chance to win and black is struggling to make a comeback.

"16. ... Rxd5 seizes the open d file."  Taking a file is only worth it if you can keep it and use to to invade on the 7th rank (sometimes an invasion on the 6th or 8th is useful to). In this case, there's no way to keep control of the file as you saw in the game.

18. ... Rc8 is a mistake, but 18. ... Re8 won't save the game either. 19. Bf4 Re6 20. Rxd6 Rxd6 21. Bxd6 Kxd6 22. a4 leaves white with a winning K and pawn ending. The extra pawn is huge in this posiottion. (Even more precise for white is 20. a4 and trading the pieces next since black has no productive moves as a resul tof the pin). 18. ... Ke6 might have been a chance to do better, but white's extra pawn will still be the most important factor in the upcoming endgame.

"7. ... d5. Looking to open up the center even more" Why are you trying to open the center? This move seems to just give away a pawn."

I think at the time I expected to be able to win the pawn back pretty easily since it was extended pretty far out and would be difficult to support. I probably thought I'd be able to force a rook onto his 7th rank after doubling them up and get even that way.  Somewhere between idea and execution it failed miserably. ;)

3...Nxd4? - "I decide to try out something a little different, which may well be unsound but was aggressive."
 

While the initial move may seem aggressive, after 5.Qxd4 White has a central queen and pawn. His position is more aggressive. If you had looked ahead and seen this you would probably have chosen 3...exd4 4.Nxd4 instead.

5...c5?! - This was a strategic error, but it had little affect on the game. Still, notice how after playing this you have a backwards pawn on d7. That pawn also happens to be on a half-open file... which is very dangerous. You also have a hole on d6 that enemy pieces perhaps could use.

7...d5? - "Looking to open up the center even more" - From a strategic perspective this is a mistake because your opponent is more developed than you. You want to keep the position closed. From a tactical perspective the d5-pawn has three attackers and two defenders so you will lose a pawn.

9...Qxd5 - "I'm down a pawn and his center pawn is advanced to d5, but chances are still fairly even." - You correctly assessed that White has two advantages  and you didn't list any for Black. I think you have to be fair and say White has more chances of winning, though you definitely still have drawing chances.

12...Kxd7 - "Although I'd like to castle, connecting the rooks by retaking on d7 with the King is a decent consolation" - Yes, that is very astute! Castling is not so important... perhaps even a mistake... when most of the pieces are already traded off. Your king doesn't need (much) safety. He needs to help to prevent the enemy rooks from penetrating into your position in the ensuing endgame.  :)

13...O-O - "Takes the open file" - Excellent thinking! If you keep playing principled moves like this I have hopes that you can draw this game.

14...Re5 - "Looks to win back the pawn on d5" - I think this is premature. You always have to ask, what will my opponent do? 14...Re5, 15...Rxd5 is too easy to stop with c4. So perhaps b5, preventing c4, is a slight improvement giving your opponent more chances to go wrong. Besides, it seizes queenside space.

16...Rxd5? - White's d-pawn happened to be sheltering your king.

17...Rxd1 - "With the more active king, I decided to accept the swap." - You are behind, so swapping pieces is NOT in your best interests. The real reason to accept the swap is because you have no good alternatives.

18...Rc8?? - Ouch! It happens... live and learn (more tactics).  :)