Looks like they have netting where the two plywood meet.Looks really pretty!
But I'm a little concerned about that plywood that's making a sort of inner-roof: I expect the chickens will try to fly up to sit on the edge of it, and walk around on top of it. So then they will be pooping on top of it, and you'll have no way to clean up there. They probably won't get stuck up there, but any chicken that has a problem might hide up there where you cannot easily see or reach them. Also, hens might try to lay eggs up there.
(It might be fine. I'm just saying what I'm thinking, so you can keep an eye out and see whether it happens.)
I'm sure the chickens and cats could get used to each other, especially if they are in separate sections.
If you are going to shut the cats in at night, do make sure they have a litterbox, water dish, and food. (It's probably obvious, but I'm saying it because if you're used to outdoor cats, you may not be in the habit of considering litterbox/bathroom needs.)