Welcome to BYC!
I live in a similar area with coyotes, rattlers, bobcats, mountain lions, hawks....we got it all! LOL
Snakes can not dig underground. And generally rattlers do not climb either. They are too heavy. Other snakes can slither up posts and such, but rarely do they climb up trees. So your snake worries are can they slither thru a hole. An adult rattler cannot get thru 1/2 inch or less. Baby rattlers can. Other snakes can slither places as well.
As long as your coop is built like fort knox, snakes, coyotes, owls, hawks and the like will never be able to enter. As for your run, you can bury fencing, railroad ties, (these are my favorite) at least a foot down. I like to use large rocks around the outer perimeter to deter any digging from raccoons, skunks, and other night time varmints. Use 1/2 inch hardware cloth on the outside of the run. (chicken wire is not only too flimsy, but rattlers can get thru it. I had this trouble with rattlers in one of my quail aviaries. Rock Rattler came in thru chicken wire.)
Build it, build over and build again. Nothing can get in if you outsmart them first.
