No flies in the coop, but quite a few in the run when its muddy. I put down hay and grass cuttings and occasionally rake it out.
Just a note, from a composting perspective, hay is and grass cuttings are "green", high nitrogen, just like the manure. You might get better results from a compost "brown" -- a high-carbon material.

Chickens don't eat many, they get a tail occasionally but they're a little too fast.
Oh really, mine grab the lizards all the time. Poor things.
I keep warning the various little lizards to stay out of the chicken pen. I hope they listen.