Hot weather could definitely explain decreased production. It's been 90+ nearly every day for the last week or two where I am and egg production is down.
It's definitely a tough decision! In my experience, skittish pullets grow up to be skittish hens. Picking the friendlier pullets is one way to do it. I also sometimes make decisions based on their various endearing quirks.
When I raise chicks I usually share with a few chicken-keeping friends...
I'm not an expert, but that looks like scaly leg mites to me. I believe scaly leg mites can lead to issues with joints, so perhaps that is what is causing her strange toe. If it were my chicken, I would treat for scaly leg mites and see if symptoms improve.
If things have gone okay for the day and your coop has space for more, it shouldn't be a problem to have her in the coop at night. If you can, it might be good to keep an ear out for bad fighting as they settle into their roosts for the night, but some jostling over the "best" roost is normal.
It's possible the eye is badly swollen and not ruptured, a friend had a hen whose eye was pecked badly by a flockmate; we thought the eye was gone but it was just badly swollen. Even if she does lose an eye, she can still be okay without it as long as you can keep infection at bay
@Eggcessive...
Here are some additional resources:
https://www.tillysnest.com/2013/12/shock-in-backyard-chickens-html/
https://poultrydvm.com/condition/predator-attack
I had a chicken survive a head wound that was deep enough that I could see her skull, it's possible for her to make it.
I'm wishing her a swift recovery.
I'm not an expert, but I don't know that you need gauze. If you can isolate her somewhere clean and the wound isn't bleeding uncontrollably, it might be best to let it dry out.
Chickens are very resilient and heal from very serious wounds. It's good that you've cleaned the wound. If you haven't already, you should isolate her, ideally somewhere warm where you can keep a good eye on her. After an injury, chickens will go into shock, so it's important to keep her...