@SmiYa0126 has give you some great guidance. Treating the eye with the Veterycin is good. I'd keep up with that, as the lesions around the eye are really the main concern with her pox. If you have a lot of mosquitoes, it's very likely it's pox.
As for the bullying, there is help for that without rehoming the two baddies. Here's my article on treating the victim of bullying so they will learn to stand up to the bullies.
https://www.backyardchickens.com/articles/chicken-bully-chicken-victim-a-two-sided-issue.73923/ This system works in almost every case.
Your run might need modification. Additional perches so the victim can have escapes helps. Expanding the run may be necessary. Minimum of ten square feet per chicken is required to minimize aggression.
As to leaving the coop open into a secure run, there is no reason not to as long as the run is covered and locked so predators can't gain entry. Mice will find their way in, no matter what. Mice run wild in my runs and coops at night, I find their foot prints. The chickens are not upset in the least with this mini-rodent nightly partying. I even had a pack rat try to construct a nest each night in a nest box, and the chickens were not bothered in the least by it. I was, though, and succeeded finally in trapping the thing.
