I don't know a lot about SF gardening, but 8 cucumber plants in a single SF seems like a lot to me. Same with 9 bush bean plants...those things spread out quite a bit more than a single SF area when mature. And as mentioned by another poster, I'd be worried about the Squash plants shading out the lettuce as they mature.
For your tomatoes, since they are all in a row, you might consider a Florida weave setup rather than individual stakes. I, admittedly, have never tried it, but it seems to be a good way to go to limit the amount of pruning, increase production, and still not take up too much space (i.e. a cage).
As for soil, the combo I used was vermiculite, peat moss, cow manure compost, mushroom compost, and potting soil.
You're probably not all that late for your area. You could have started the seedlings inside to get a jump on the growing season, but it's probably still too early to put seedlings and direct seed into the ground for DC. Tax day is the standard around Charlotte, but you guys have a couple more weeks of worrisome weather up that way.
As for watering, I recommend a watering wand if you're going to water by hose. Preferably one with an adjustable flow, as you need to turn the pressure down when the seedlings are first coming up. Also, be sure to keep the water off the leaves of the tomato plants. Once the plants are mature, I don't use the wand at all and just use my thumb to control the flow and water at the base of the plant.
The good news about a raised bed is that they tend to drain well, so over watering is hard to do.