It's winter. They need the layer of fat to keep warm. The feed sounds good as long as you're not adding much of the 16% protein feed. I like Bar Ale, but I'm biased since that feed is from my home town. ;)
Edit: I've also noticed that older quail have more fat.
I don't think it will hurt to try the antibiotics.
Three years old is old for a coturnix. Her age and being alone may be factors in her health condition, though it sounds like you're treating her quite well. Even if she doesn't use hiding places, it's good to have them available for her just...
That looks like the second eyelid. I would start with a saline solution rinse to keep it clean.
How old is she?
How many quail do you have?
What is their home like? Bedding? How much space? Hiding places?
Is she eating and drinking normally?
Does her poop look normal?