I’m in AZ and we get really hot too. There is corrugated roofing that’s fiberglass I think. You can get it in clear or white. That has lasted a long time for me and doesn’t get hot like metal.
Just thought of that, I have it on my coop roof.
I wouldn’t do metal, it does absorb a lot of heat and will reflect that into the run. Wood chips and straw are excellent for the mud. They will improve the soil structure and you’ll have less of a mess, but will still need to add to it on occasion.