Game analysis request

Based on what I am seeing your biggest error isn't so much that you took so many of those exchanges, it's that you let Black make the first move. As the game opener you need to be on the offensive and if anyone is offering exchanges it should be you. He managed to push the game because you were too timid early in the game and fell for easy gratification.

You are correct, I should have played in a different manner. Perhaps 6.Nxd4 instead of Qxd4 in an effort to keep more pieces on the board. Or simply a different opening choice.

Fast forwarding to 20 where you two were even you continued to let your opponent guide your movement. At that point your next move should have been to move your knight to D4 instead of moving a pawn in order to attack his side of the board affecting his ranks. Given, his next logical move is knight to E4 for check but you stay in control by moving Ke3. He has three options: Back off to a neutral position, attack a pawn, or stick around. Because this is his last attack piece he will back off to Nc5 or Nf5. Now you march your pawns and build an attack.

Yes I definitely wanted to play more aggressively with 21.Nd4. However I was concerned with his knight going to g4, forking my f2 and h2 pawns.

The 4 Knights opening doesn't give a whole lot of advantage, and the position after move 13 is pretty typical.  You have a very small advantage due to slightly more active piece placement.  It isn't much, but if you are very careful, it may give you chances to win without much chance to lose.  See the 14 games here:  http://www.365chess.com/opening.php?m=27&n=51172&ms=e4.e5.Nf3.Nc6.Nc3.Nf6.Bb5.d6.d4.exd4.Qxd4.Bd7.Bxc6.Bxc6.Bg5.Be7.O-O-O.O-O.Rhe1.Nd7.Bxe7.Qxe7.Nd5.Bxd5.exd5.Qf6&ns=3.5.5.6.47.57.1096.4992.10020.11776.25637.31356.47381.13507.12647.13508.15425.15902.15426.45282.48986.49735.48987.49736.51033.51172.

Thanks for that link. A lot of those master games followed similar roads as my game and some of them were able to squeeze a small advantage out of the position.

Stockfish rates the position at +.2 for white, with 14. qe4 being the best move.

Qe4 makes a lot of sense. I was worried about him contesting the e-file and equalizing if I moved my queen away but in this case at least I control the file, for whatever that's worth.

You played really well imho, you just selected a boring opening and your opponent played it well enough for it to lead to a miniscule edge.  6. nxd4 is the main move, and leads to a slightly more dynamic position, but 6. qxp is sometimes played and wasn't an error.  

The small chess club I "grew up" in had a senior master that exclusively played the 4 Knights.  There is nothing much wrong with it if you are willing to try and take a very small advantage and nurse the heck out of it.  However, it is not necessarily the best way to avoid any opening study, since if you don't know the lines it can be difficult to get any edge at all. 

Yeah, it's difficult to gain an advantage in the four knights if your opponent plays well. The amount of theory in an opening like the ruy lopez for example, is quite daunting. I like the four knights because a lot of the theoretical moves are quite natural and intuitive to play even without memorizing the theory. I was inspired by old Capablanca and Nimzowitsch games to play this opening. However they had to skill to convert a win from even the tiniest of advantages, and I don' t have anywhere near the level of technique required to accomplish that. I should look into switching openings.

29. f5! was your missed opportunity to get a real advantage per Stockfish.  You played 28 moves without a real error, so congratulations.  If you want to learn from the game, play out some lines with the computer and see if you can understand why f5 was a play for a real advantage.  (Of course, you could have played f5 on move 31 to the same effect.)

So yes, you should not have taken the draw.  29. or 31. f5 gave a real chance to play for a win.  And this is the exact sort of opportunity you have to be excited about if you are going to play a relatively quiet opening like the 4 knights.  

Yes, f5 looks quite strong, a pity I missed it. Honestly, I think one of my biggest mistakes in this game was my evaluation of the endgame. I thought it was a completely dead draw but white is in fact pushing for a small advantage. It may not be enough to win but I should have at least challenged my opponent to find the accurate defence.

Possible improvements are 6. Nxd4, or 8. b3 (intending a fianchetto), or on move 10. (or even 11.) h4.

Yeah those suggestions make a lot of sense.