You never "need" to add a light to the coop, although it should have some natural light, so that they see when it is getting dark or light outside. They do indeed sleep better when it is dark. Just sit outside the coop for a night with a light on, then without, and see how much restlessness...
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I would go ahead and do it even though you are doing it late. Worth a try, and actually I suspect all will be OK as long as it is removed before cooking.
There are any number of possibilities.
If they are over a year and you have never given a commercial wormer you could start with that. I would recommend ordering Roosrer Booster from My Pet Chicken, here. You won't be able to see most worms in the poop. You could also tke a poop sample to...
It would help if you would tell us your general location in your postbit; you can add this in your profile.
If it is hot where you are, chances are it is cooler where they choose to rest than n the coop. In hot weather they need lots of breeze and shade as they don't tolerate heat well at...
Dig a small drainage ditch around the pen with anorger draining ditch going downhll from there, maybe?.
Here is a great article on fixing a muddy run; perhaps something will help.
https://www.backyardchickens.com/a/how-to-fix-a-muddy-run-chicken-coop
I would not personally limit chickens to that amount of space, especially in winter. I would say it is marginal. A larger outdoor area that is free enough of snow that they will use it in snowy weather would probably be the first thing I would add.
Here is an excellent article on space...
I wonder if those are tapeworm "bits.". They look a bit like grains of rice. Most worms are not visible to the naked eye in the poop -- the only other one I've seen in a pic of poop is roundworms, which ussally happens only in a heavy infestation, and those are not roundworms (which are thin...