I covered our run with inexpensive chicken wire, stapled securely, and then put a large, heavy duty tarp on top of the wire. Slope the tarp so that rain water doesn't collect in it. Learned that the hard way. :). The tarp keeps out rain, sun and prevents flying predators from seeing the birds and getting stuck in the chicken wire if they decide to fly in for lunch or dinner. A predator can get through a tarp with no trouble at all, but has a harder time getting through chicken wire, which is why we use both.
We also put an old large sign and some 4 x 8 plywood on one side of the roof to help with rain, and we're going to use some old doors as "walls" on the outside of the run - just further protection from elements. It won't be beautiful, but it's not like the birds care, and it will be inexpensive and do the job.
What are you securing the screen door with? We have two hook and eye latches on our door (top and bottom), but the lower one has a security feature (I guess for little kids) where you have to slide back a mechanism before you can lock or unlock it. They're uber cheap. Have you covered your screen door with wire? Regular screening fabric won't keep a predator out. But you may have already planned for that.
Sorry if you've already thought about all this.
We left ONE weakness in the coop roof - about a four inch venting area - and we stood back and said, "There's NO way a raccoon is going to be smart enough to climb up and in through there" and of course, that's exactly where he came in and took the first chicken. And he was smart enough to climb out of the run/coop the same way he climbed in, too. It's now secured, but we're down one lovely chicken. Poor thing. We've learned not to underestimate a hungry predator.
You've got a nice looking run. Good luck with outfitting it and getting it going.