I’ve never heard of testing for diseases weekly. That would be unusual and very expensive. Some people who are NPIP tested will get testing once or more a year.That testing is just for pullorum, although some may offer testing for certain respiratory diseases.
I would make sure they have a lot of thick bedding in the coop, and block any windy areas, but make sure they still have overhead ventliation.
You are correct that the nictitating membrane or third eyelid can partially cover the eyeball. Also when they close their eyes, it can look like they are white.