Coccidiosis can come from the soil, it is naturally occurring. Are they having bloody stool? If it is coccidosis they die pretty quickly without treatment (within a day or two)
Umm picking off that black area that is on the side is a tough call my experience with pealing off dead hardened skin was with this hen pictured. This is the rear of the chicken, and I peeled off the piece in this picture and it reopened the wound and started bleeding again. The other ones that...
They can rip them out and often times they will bleed a lot, so you can dab it with iodine or antibiotic oinment to stop bleeding and stop an infection. They will be fine without a nail, if they are getting long and not getting "trimmed" by them scratching you can clip them.
I have had to nurse injured chickens a few times in the last several months twice from dogs and once from a raccoon attack. I made sure they had electrolytes in their water and feed/gave water with a dropper the first several days. I also slathered antibiotic ointment on the wounds and kept them...