All the advice you've gotten so far is sound, but I just want to clear a few things up for you just so you know.


Rims
The inside distance between the rear dropouts is 124 mm. The inside distance between the front dropouts is 100 mm. What size Velocity Deep-V rims are most suitable for my frame? 700C? I know I want MSW because I want to use brakes.

I'm not sure if this is what you're implying, but the spacing of the dropouts on your frame have nothing to do with rim size, that is an issue for the hubs. Most of the flip-flop hubs you're going to be looking at will have the right 100mm spacing for your fork, and can be easily made to work with your rear dropout spacing with a couple of washers (track hub spacing for the rear is normally 120mm)
700c rims are going to be your best choice. You'll have a much better tire selection. If your frame was made for 27" wheels, you may have to find some long-reach brakes, which isn't too difficult - just search this forum and you should be able to find plenty of info about that.

Hubs
Assuming I'm going to run 24 spokes in the front and 32 spokes in the rear means I can't use a Surly hub for the front. A White Industries H2 hub was suggested to me. I can still use a flip-flop Surly hub in the rear.

If it were me, I wouldn't bother mis-matching hubs just to save a few grams on spokes unless I was building a serious racing wheelset. Deep Vs are known for their ruggedness and looks, not light weight.

Spokes
I'm at a loss. I know that DT Swiss and Sapim are the most common brands. Which model of either would you guys suggest? Is a radial lacing pattern for both wheels the best bet? I was told that I should run 24 spokes in the front and 32 spokes in the rear for if I were to run 32 both ways the ride would be harsh. That and most of my weight is going to be on the rear wheel so logically it doesn't make sense to have wheels with identical spoke counts.

Running 32 on both wheels won't be a problem as far as harshness goes. One of the main benefits of using a radial lacing pattern is that it results in a stiffer wheel, which would make your ride more 'harsh'. Radial lacing is fine with me, but I'd rather just run 3x on both wheels - I like being comfortable giving my wheels a beating. As someone said above, radial lacing on the rear wheel is not doable. Some people will get away with doing a radial patter on the non-drive side of the rear wheel, but you don't really need to bother with this.

I'm 6'0", 155 lbs, which are figures I know can be significant in the wheel building process. I went onto Velocity's website to check out some of their online retailers. ProWheelBuilder.com could build a wheelset like that above with DT Swiss Revolution 14g spokes for slightly more than $350. Is this relatively cheap? Who is the cheapest, most reliable Deep-V wheelset builder I could call or email?

Depending on what colors you want, Ben's Cycle has some cheaper wheelsets here.



Edit: or I could have waited five minutes and let everyone give all the answers individually. Oh well.