Welcome to the AZ thread, and to BYC!
We just used some regular wood from Home Depot and lots of wood and metal scraps from refurbishing our house. Like others have noted, shade and ventilation are of upmost importance here. Be sure to keep in mind that chicken wire is just for keeping chickens IN, but it doesn't keep predators out. Some predators, like coyote and kit fox, will dig to get a meal f possible. Your use of hardware cloth is a good idea. In the hottest months, shallow pools of water or mud puddles are good. We have a roof over the run because there wasn't enough shade. Here are some photos of our coop and run, just for an idea. We are working on upgrading the roof over the coop this month, and I'll post more pictures when that is complete! Good luck and feel free to keep the questions coming!
Your run is how is basically how I was planning on doing mine. I can't go over 6.5 ft tall due to the HOA though. Did you just use shade cloth on the sides of your run?