In my experience, they like full sun and plenty of water. I've had the best success in my improved/raised bed (but I have otherwise very dense and poor draining soil.) Yes they will grow on a trellis or fence and that is a good way to save space. Watermelons really like to spread out, just like pumpkin. Are squash vine borers a problem in your area?

My family always grew 20-40 acres of watermelon each year when I was a teenager, which has been a few decades now. We would always plant custom greenhouse grown plants that were about 4 to 6 inches tall, so I can't help on the from seed part. (I don't remember exactly how many, but it was measured in 32 ft trailer loads, usually 2 or 3 per year, from the greenhouse nearly 200 miles away) I can tell you that we grew them with no irrigation with the plants planted in rows of black plastic mulch (trash bag material), the best melon production was in years with moderate spring rains, excess rain caused inferior melon production or failed crops. This was in SW Louisiana, areas with sandy soil generally had the best production, most of our land was fine sandy loam soil, however the best melons were always from the more sandy regions. As to fertilizer I remember we always added a lot of extra potassium when preparing the soil in the watermelon patch each year.

What are "squash vine borers"???

Squash vine borers don't attack watermelons. Melons need full all day sun and like loose, preferably sandy soil. They need water, while the plants are establishing themselves, but little after they set fruit. I till my soil every year and add whatever nutrients, a soil analysis says, when I till. When the plants start to run, I give them an extra shot of nitrogen, and nothing after wards. After the plants have set fruit and they start to grow, if see them starting to wilt in the afternoon sun, they need irrigation, but only do it in the early morning. If you water them later than that, the melons will have a tendency to split, due to the excess pressure from the water. I also direct seed mine, with permeable black plastic in between the rows, in order to retard weeds.

How much soil is needed for a watermelon? Can I make a small 1'x1'x1' raised bed box per watermelon seed? Do I need something larger?

1x1x1 meters? Maybe

The roots run in the same directions as the vines do. You're asking the plant to do an awful lot in a real small space, so don't expect much in return.

I am planting water melons too, in zone 6b. Since our season isn't long, I started them indoors in milk jugs. Can I harden them off and plant outside now, or as soon as our frost danger is past? And what do you think of the idea of planting them on the mound of half-decomposed compost? Too much organic material??

Watermelons need heat and full sun. I wouldn't plant the starts, until your soil temperature is consistently in the 70's. They should do well in the compost mound.

Watermelon doesn't like being transplanted.

best bet is make a small greenhouse tent over the spot or use black plastic to heat the soil up sooner than it would otherwise.

Watermelon does not like to get transplanted? I am also new to growing so guess I will find out how it goes. I repotted once due to leggy seedling and will plant out/transplant when it gets warm.

This same plant as of a few days ago with newly sprouted seedlings