My compost pile is near the run. I cleaned the bedding from the chick brooder and added it to my backyard pile that heated up QUICK with all the nitrogen. 4 pallets and zip ties. QED.
Yes. I am now throwing kitchen scraps into the run along with yard waste and wood chips from the collection point. Lots of nitrogen coming out of the birds, so LOTS of wood chips to soak it all up. Cardboard boxes, grass, leaves, sticks, twigs, chips, logs and such.
Some people just scrape the bottom of the coop out into the run and let the chickens mix and composting happen there. Rinse, dry and add fresh absorbent to the coop. Repeat as needed.
There are several lists of poisons to chickens that you can search for. Pothos is the one at the top of my mind that is poisonous to everything and invasive. Pothos should be burned and buried. Pretty houseplant. Don't let it outside after midnight...
I keep a separate pile for paper composting. Don't really have a good reason. Just don't want the paper blowing around the coop. Recycle what we can.