I don't really know what I'm talking about, but I've never let that stop me before...
I've generally seen these sorts of structures with some sort of opaque roof on the top and the diagonal parts. Most often rigid, but sometimes a tarp. Mostly for the shade but also to help shed rain.
I suspect if you start with tarps that sooner or later you'll want something rigid. But maybe not.
It won't take long for any grassy area within your run to become a wasteland, unless you plan on tractoring that whole thing around a fair bit, is the impression I get. But that's the sort of thing you're likely to discover on your own.
I feel like you might want a kind of short retaining-wall all around either the inside or the outside, partially to keep in litter but mostly to stop the whole thing from taking flight on a windy day.
I don't think you'd strictly need them, but you might want to put up some sort of walls in a corner or two so they can get out of the sun/wind/rain or whatnot. If they can just pop back into the coop whenever they feel like it, that would probably be sufficient.
You're planning on a predator apron, right?