Definitely keep the cukes to the sunniest part of that bed. I've tried cukes in less than full sun a few times and they seem to need all day sun for any health and productivity. Melons are like that too.
Chard, lettuce, fennel, parsley, beets and celery all do OK in high filtered shade for me. (Tree's thinned)
You can make the situation a bit better by thinning the trees causing the shade. It doesn't have to be a major production either. Just thinning out the smaller diameter branches coming off the big ones can make a big difference.
Oh and I would plant that whole slope behind the fenced area with cukes and melons and zucchini and interplant them with some pretty tansies and marigolds! To deter some of the pests. The bambi's around here ignore my cucumbers so no need to fence. Plus that slope is probably annoying to mow.
I wouldn't even worry about tilling the whole thing. Just chop some 2x2 holes in the sod, make some mounds and plant!
Stacy (aka she who is slowing removing lawn and planting other stuff)