Welcome to the forum
.
To answer your questions:
1. No, there is no medical test. People from certain countries must get a TB test, but that doesn't apply to the US. You pay the £600 IHS surcharge at the time of visa application (the visa will not be processed unless you have paid it) and that's it.
2. I would have thought there would be treatment for long-term dementia care, considering the NHS covers all medical needs.
3. I don't know anything about residential care eligibility either in Scotland or England, I'm afraid. I believe the Local Authority will assess whether/what type of care is needed and then look at your capital to determine whether you qualify for financial help. Doing a quick Google search, it doesn't mention whether they look at overseas assets or not.
4. Well, NHS Scotland is a separate entity from NHS England, so it may have some differences in how care is provided. In Scotland, all prescriptions are free to everyone, whereas in England, prescriptions are free to over 60s and under 19s (if they are in full-time education), and are free to certain groups of people, but everyone else has to pay for prescriptions (£8.60 per item).