Another idea is if at all possible take your grass clippings and spread that out in the run to help keep them off the muddy ground and absorb some of the water
I agree with your guesses 100% also and I do not suspect mites based off of the photos dust bathes are good but diatomaceous earth can be sprinkled on them if the symptoms of mites become more visible
I also recommend using craigslist because many of the coops or play sets can easily be used i know someone who built a coop by taking a swing set and removed the swings ect. And just put a door and chicken wire around the whole thing and it turned out looking half decent also a tip is to allow...