In e4 e5 openings, if I get a chance to safely, I would play d5 as soon as possible, trying to contest his e4 pawn i.e. move 3 here. Not only do you strike at his centre but you wont block in your own bishop as you do after 4...d6, and development is freer.
You didn't blow him off the board when he wasted moves because you did too. 3...c6, moving the knight twice (because you had taken away the best square for it with c6) etc.
In e4 e5 openings, if I get a chance to safely, I would play d5 as soon as possible, trying to contest his e4 pawn i.e. move 3 here. Not only do you strike at his centre but you wont block in your own bishop as you do after 4...d6, and development is freer.
You didn't blow him off the board when he wasted moves because you did too. 3...c6, moving the knight twice (because you had taken away the best square for it with c6) etc.
I didn't need c6 to support the d5 break, thus challenging his e4-pawn? Can you type an alternate continuation, how you would have played it after the first two or three moves? Thanks. 
You would have played it this way?
Yeah precisely like that. Bear in mind I'm no master but black's game looks easy there, and should have no development problems.
You don't need c6 to support the d5 break, just if you want to keep a pawn on e5 and d5 which I don't think is a realistic ambition as black.
First let me say that what white did essentially only ceded equality. You shouldn't beat yourself up over a draw, although I know how it can feel when your opponent plays moves you know in your heart are not best, and you really feel you deserve to win (but this is a dangerous notion with so much chess left to be played).
Ok so 2.a3 wastes time. I was looking at 3...d5. Yes, also because I play a similar line as white so pattern recognition helps, but you can also tell yourself opening lines is good when ahead in development (and of course winning if significantly ahead).
3...c6 prepares d5 and d5 is good. The downside to c6 is it takes that square away from your knight, so you're essentially planning less aggressive development as well. The trick to punishing non-development by your opponent is almost never immediate. Many times it's as simple as bringing out all your pieces as fast as possible and then working to open lines... the punishment comes later.
Notice after 5.h3 that white has 2 pieces developed to your 1 and center space is equal. c6 means you coudln't protect your e5 pawn with a knight on c6 so you had to make another pawn move d6 and now white has his half move advantage in development again.
6...Na6
Hmm, aren't you preparing d5? Doesn't Nbd7 instead of Na6 allow you to play d5 on the next move. i.e. the pawn advance d5 takes away a defender from your e pawn, and Nbd7 preps d5 by defending the e pawn.
10.b4
Your available break 10...a5 is almost always a good break when you're developed. If you can get rid of a rook pawn (the least valuable pawn in middlegames) and open your rook this is good. White also does this same break in the ruy. After black plays a6 b5 white often finds a4 is a good move (so again, a pattern to remember). Your move h6 is fine, this is just so you know, it's almost always something to consider.
12 d4.
yay, scope for your f6 bishop. You respond by capturing the pawn, good.
13...Be7
boo, no more scope for your bishop.
14...d5
yay, opening lines.
15...f6
yay, opening lines. Is the king safe on the light squares? d5 looks solid, so that diagonal is safe. Not many attackers around black's king. Ok looks good.
24...Nb5
I liked your knight and bishop better than their counterparts, but after some trades you go into the very drawish opposite color bishop endgame. This is where any remnants of an advantage disappear forever. Not an uncommon error, I'm working on my endgame transitions too actually, trying to notice when my advantage carries over or not. Supposedly all amateurs suck at this lol.
So to recap, development beats non-development (not pawn moves). And only after you're ahead in development look for pawn breaks (although you could have done this as early as move 3).
Knight on d7 supported your idea better than Nc7 I believe because what was needed was not more control of d5 but a defender for e5.
Your center breaks d5 and f6 were good. Your transition into the endgame was not good.
Hope that helps :)
So do you think 3...d5 is best waffle?
But yeah, 6...Na6 doesn't fit in with your plan of preparing d5. With the knight on f3, the e pawn need supporting in order to advance your d pawn.
In general in these king pawn openings, white is trying to establish a pawn centre of e4 d4 and black wants to play d5 (when it works tactically) to strike at this centre.
Black may have other pseudo transpositions, but yes, I'd play 3...d5 without much thought... which is unfortunate because I don't really know that reversed scotch thing... not that'd I'd be too uncomfortable.
On looing at that I may try Be7, just a calm developing move that asks white why he thinks a3 was useful in any way. Be7, Nc6, 0-0 and black's a tempo up in whatever they get into. White can play d4 but it's not a big deal if black is castled and white isn't. If white doesn't (or even if he does) black can hit back with d5 (notice in 1 move, instead of d6, d5). So actually I like this a bit better I think, although it's totally a matter of preference. I don't know the scotch so...
You don't win a game because white used 2 tempo's. Your own plan with d6, c6 and the knight hop was also very slow.
I don't like how you opened the e-file with all the pieces on it at all. They looked vulnerable. You could have had something with Bf6: Instead the knight was hanging and you exchanged queens.
Endgame is fairly even, and the doubled pawn was no good for you.
2.a3 is just giving back the move, nothing essential here. After 3...d5, Black is playing a Scotch, with white having the extra and slightly irrelevant move a6/(a3) in. So this is still a game...
You couldn't take avantage of his slow development because you didn't open the position fast enough. c6/d6 is not that fast and doesn't develop pieces either. Anyway, not a bad game at all : neither you nor your opponent blundered anything.
1. 3...c6 prepares d5 and d5 is good. The downside to c6 is it takes that square away from your knight, so you're essentially planning less aggressive development as well. The trick to punishing non-development by your opponent is almost never immediate. Many times it's as simple as bringing out all your pieces as fast as possible and then working to open lines... the punishment comes later.
2. 6...Na6
Hmm, aren't you preparing d5? Doesn't Nbd7 instead of Na6 allow you to play d5 on the next move. i.e. the pawn advance d5 takes away a defender from your e pawn, and Nbd7 preps d5 by defending the e pawn.
Thanks to everyone for the analysis. Wow! Most instructive.
What was I thinking in this game? 
1. My plan after White played 2.a3?! was to play a reversed Italian Game: Classical Variation, Giuoco Pianissimo. Not for the piece placement, but for the pawn structure of e5, d6, c6...with a d6-d5. I remember the IM's here talking about the pawn structure defining the piece placement. My pieces were not placed according to the pawn structure, it seems.
The other reason why I played ...c7-c6 was to open the d8-a5 diagonal for my queen. It's a diagonal I'd like to explore, and Nc6 doesn't give me that opportunity, I guess.
Reversed openings happen often in my games because White does not place two pawns in the center that often, sometimes choosing to fianchetto both bishop, attacking my ideal center from the flanks.
2. Looks like I was over defending the d5 square. The idea of Na6-Nc7 was to add a defender to the d5 square, so that when I play ...d5, that pawn will be well defended, and I won't need to worry about losing the exchange on that square.
Also, I was so focused on this d5 break that the move ...Nd7 never entered my mind. I ended up struggling to defend e5, and that slowed me down.
I hate people when they lose they go away
My opponent makes two time wasting rook-pawn moves in the opening, I guess to prevent my bishops from pinning his knights, or preventing my knights from hopping two ranks forward. So I think to myself, "this guy doesn't know how to play the opening".
My plan after c7-c6 was to break a few moves ahead with d6-d5, opening the center, and then using my better development to blow this guy off the board. It never happened. Where did I go wrong?
I did move my queen-knight twice in the opening, but that was part of my plan, redeploying it to a more useful square after c7-c6 with Na6 to Nc7, supporting the d6-d5 break move.
My opponent refused a draw, even with a bishop of opposite color ending, and it took some time to get the server to call it. I must not have clicked on the draw button at the right times.