...what varieties of chicken would do best for this. It would be rather unfortunate if I put in this effort only to find out that the chickens couldn't get to the coop. I have a friend who has spent a lot of time on farms and assures me this will work with some chicken breeds but I have not been able to find any examples of someone actually doing this.
For chicken types, I would say not these:
silkie-type feathers or frizzle-type feathers
big, heavy birds (over about 5-6 lbs expected weight)
birds that cannot see well because of feathers (some crested breeds, a few with large muffs)
That leaves you to with most small breeds and bantam breeds as possibilities.
Since you also want them for eggs, I would try Leghorns or Hamburgs. Both breeds come in many colors. The Hamburgs will tend to be smaller, and lay smaller eggs, than the Leghorns. Of the Leghorns, the White variety probably lays more eggs than the other color varieties, because it's the one strongly selected for commercial use.
Some bantams would fly even better, especially the small slender ones like Old English Game and Sebright, but they will lay fewer eggs and smaller eggs. I've had Old English Game Bantams that could fly straight up from the ground to a tree branch that was about the height you're talking about. I suspect they could have gone higher, but never saw it to know for sure.