During math class today (I'm such a good student) I was wondering about 4 Qd3 h6 5 Bxf6!? Nxf6 6 e4 (getting the break in before Black prevents it with ...c6/...Qa5; i.e. 5 Bh4(?!) c6! when 6 e4? loses to Noah's Ark-esque tactics) 6...de 7 Nxe4 Nxe4 8 Qxe4, although admittedly I haven't really looked at this seriously at all yet. Sure the position's relatively simplified and Black has the b-pair, but I was thinking I could stop ...0-0 with the Bd3/Qe4 battery and try to hold up ...c5 and ...e5 with some sort of d-file pressure. From what I can tell, it's fairly unexplored (which is what I'm looking for in a surprise weapon), and I don't think White is worse (White doesn't seem to have lost this position in practice yet, although of course that stat isn't so useful), although the question is if I can play for a win from such a position..

No, it's not. 

Hi all I am playing the Veresov in correspondence in ICCF Veterans World Cup World Cup and
ICCF Diamond Jubilee World Cup. Obviously I cannot post on the forum about games that are in progress but I will say that the Qd3
Lines can be just as much down the rabbit hole as the f3 lines.

Having studied very carefully Veresov with Qd3 I came to a conclusion that the most appealing answer for black seems to be the setup with g7-g6. No matter how I tried to find even a tiny edge for white...it was all in vain. Black has a comfortable setup and counterattacks very successively on the queen's side where white castles long. So I stuck to the positions where white is developping slowly after Bc1-g5, e2-e3 and Bf1-d3...treating the position just like I play Chigorin with black but with a move up. In the main line of this setup white is trying to push e3-e4 no matter what...but it gives only a full equality, so I rediscovered an idea with a2-a4-a5 (instead of e3-e4) against a7-b6-c5-Bb7 formation in the critical position of this variation. The last corr game of mine (not finished yet) in which I executed this plan will bring me a full point very soon. 
So, my conclusion is that white can obtain a small edge only in the main line.  

Of course now there's the book A Ferocious Opening Repertoirewhich I think covers both f3 and Qd3 mentioned above.  I still don't completely trust this stuff but I suppose that' not the point.  I also think it could be helpful to compare some of the positions from 1.d4 d5 2.Nf3 Nc6 from Christoph Scheerer's  (Wisnewski at the time ) Play 1...Nc6!.

@CraigEvans
"It's a fairly major switch to be going from 1.e4 to 1.d4"
If you ever seriously consider this: after 1.d4 d5 2.c4 White can play in KG-style, as soon as Black plays dxc4. No details here, as this is way off-topic. Something for a new thread?

What could be more appealing than a suicidal move like 6.g4? This is real chess!!

3...Bf5 doesn't worry me, 4.f3 looks playable, where 4...c5 looks like the only move which should cause concern to white (I wonder if white can then continue in a sort of albinesque way with 5.e4 anyway?!). Otherwise something like 4...e6 5.e4 dxe4 6.g4 is taking us back towards BDG lines which should, at the very least, be fun.
3...g6 4.Bxf6 ef 5.Qd3 intending O-O-O and a kingside pawnstorm looks fun as well - objectively at GM level it might tend towards equality, but at the 2000 level I can imagine it being a pretty good weapon. Hector certainly scores quite well with it, and gets fun games. The main point is that it removes learning against several lines. 

I might even consider playing against 1...c5 and so on in 1.d4 fashion - or even with 2.dxc5, which seems to offer reasonable chances for an edge with best play. Of course, I've been playing for 7 years and have been almost religious in my use of 1.e4 (or 1.f4 knowing that my opponent would allow a transposition into KG positions) OTB. It's a fairly major switch to be going from 1.e4 to 1.d4, but a question on how to go about this would be best served in another forum I think...