It's interesting to think of the adversarial element in #5 where vision test results are used to qualify for something. In this case a completely unsupervised test is really easy to cheat on -- people can just lean in close to the monitor! If you're not giving people something that they can use to receive a benefit like a job or a license, that incentive to cheat seems weaker, but maybe people will present their fresh prescriptions (!) as purported proof that they have very acute vision.
I was thinking more about psychological aspects where people might not want to admit that they have certain vision problems, so they might feel an incentive to convince themselves that they saw the correct thing. The order and context of presentation might affect how easy it is for people to convince themselves of that. I know I've taken similar tests in person at the optometrist (like looking at a grid to see if any portions appear distorted), but I don't remember exactly how the optometrist asked me to confirm what I'd seen.
This may be an underappreciated soft skill on the part of medical professionals -- getting people to tell the truth about their perceptions in diagnostic tests, or noticing when people may be dishonest or simply uncertain. So that may be pervasively tricky for you to address, at least with a small percentage of patients: if they want to think of themselves as having good vision, they may consciously or unconsciously fudge the results a bit so the assessment comes back better.