Howdy!
I'm thinking it might be several things going on at once.
Yes, she may be a favorite for the roosters and that would cause treading damage. 5 hens per rooster is a lot for large breeds, and dominant roos will make a point about it. (I had 4 roos on my 20 hens for a brief time. I had some hens turn up like this. so I reduced the number of roosters.)
I think some of the red could be sunburn, in addition to a possible pest. It never hurts to sprinkle a little DE around and in the nests. I add it to their dust baths, and am thinking about adding a little tobacco, just a smidge, to help with some of the pests my girls get.
The geese can be feather pullers. They are great flock guardians, so I hate to throw shade at them, but mine will get a bit "feather-pully" when they're frustrated.
The other hens will pick on her, too. And getting new feathers to grow in can be a hard thing.
My advice (which is worth what you pay for it), is to give her a safe place to run; reduce your roosters by at least one unless you really need all of them (I currently have 2 roos over 22 hens, and one will be going away soon.); put a chicken apron/saddle on her (there are several free patterns on the internet and they're easy to make - she's a medium sized hen for the purposes of the patterns); treat her skin tear with your favorite wound care (I love this blue powder that has activated
carbon and sulfa powder in it. It Stops bleeding and helps the hen to heal.); and just let her heal up. It won't take too long.
Otherwise? Maybe up her protein? The other chickens will be going after any new feather growth, so watch to make sure she's not getting too harassed.
Aside from all that, she looks like a very healthy Barred Rock. Great hens.