We did consider using the 2×4 welded wire for the run (we've used it for our past runs). But I like the idea of being able to keep wild birds and the squirrels out.
The biggest threat is going to be dogs and cats (both our own and neighborhood strays) and aerial predators. I suspect we had hawks nesting in the trees between our property and the next last year and I've heard owls for a couple years now. Other than that, I've seen a possum in the neighborhood, and while I haven't seen any raccoons I know we have them in town. We're not that far from the edge of town, I wouldn't be surprised if we get the occasional coyote or fox although I don't think they're likely. I haven't seen any rats but I know we had mice in the house when we moved in and I suspect they are currently living in the garage.