Chickens actually love warm baths. Who would have guessed?
I think it may be time for an internal exam. That poop looks like it may have cooked egg material in it, although it could just be yellow urates. The small piece of meaty material probably isn't anything good either. That is likely a small piece of her reproductive equipment. Not generally anything to worry about because that stuff sometimes happens, but in light of her other signs it is definitely something to take note of and further investigate.