Why on earth would anyone play 2. d3???
C20: Leonardis Variation and 2...d5 (?!)

I was wondering why White would play 2.d3, but yes, the KIA (opening system) does have the move 2.d3.
The opening is not a series of specific moves, but rather a system that can be played from many different move orders. Though the KIA is often reached via 1.e4 followed by d3, Nd2, Ngf3, g3, Bg2, and 0-0, it can also arise from 1. g3, 1. Nf3, or even 1. d3.
When I played the KIA, I didn't open with 1.e4, because the point, the way I learned it from a video, was to play e2-e4 last, for flexibility, a Reti approach, so 1.e4 2.d3 looked unfamiliar to me, but now that you mention it, one could get a KIA set up just fine, with a rock solid pawn chain with c2-d3-e4. Nothing really wrong with that. At least when I played in that style, I didn't lose as many e-pawns!
Edit: I forgot that Bobby Fischer would play the KIA with 1.e4. Oops! It's been a while.
The KIA can look like a Reti Opening on move one with 1.Nf3, but in this game, the opening settles into a KIA.
One of my favorite KIA games was played by Bobby Fischer, where he meets the French Defense, after 1.e4 e6, with 2.d3 (!?) and arrives at a KIA. Folks here would beat me with a stick for committing such a grievous error in opening principles, but Bobby did just fine, but then he was...Bobby Fischer.

2...d5 looks playable. I guess many players prefer to play 2...Nf6 first, so that they can recapture on d5 with their Knight.

2...d5 seems dubious since u just enter a reverse centre game, but is it so troublesome for black?
Moreover, black can play gambit style with 3...Nf6 and i agree with pfren told about it, so black will have some difficulties if it doesn't take d5 with the queen.
If u compare with centre game, white doesnt have Bc4-Bb5 stuff, and on Nc3 attacking the black queen, black can pin with Bb4.
So it seems quite playable to me, although i would prefer to take with another piece than the queen on d5.
Thus i would prefer to develop my knights before going for d5, let's say 1.e4 e5 2.d3 Nc6 3.Nf3 Nf6 and d5 next move.
I think white has some independant tries, like 3.f4 but gods of development may not be very happy with it.
After 1.e4 e5 2.d3, why is 2...d5 a mistake? The most popular reply in Game Explorer is 2...Nf6, followed by 2...c6 and 2...Nc6. It looks like White is opening with the Philidor, with colors reversed.
I'm not at all familiar with White doing that, so I treated it like a Philidor Defense and played the most popular reply if I had the White pieces, which is to push the d-pawn two squares, freeing up both diagonals for both bishops, and having the ideal pawn center.
How can that be a mistake?
This was a 10 0 Blitz game, which I was in the mood for. A litle variety after several slower games. I now see the 10 0 games as a chance to get experience with different openings, and have some fun along the way.
There were tons of mistakes made from both sides in this almost miniature that resulted in checkmate, but that is not the focus of this post. However, it would be very cool to have a faster claculating machine (brain) for these speed games, as I missed a lot of cool tactical opportunites. It didn't help at all that I failed to defend my center pawns. What a mess!
I give myself this (!!) in speed games when I follow my opponent's blunder with the best possible move.