My life, Across, that's a can of worms!
Doctors do work both for the NHS and privately; the surgeon who did my most recent exploratory heart procedure/operation does one day a week for the NHS hospital where I had it done on the NHS, and the rest of the time in a private clinic. The eye-surgeon who did my laser surgery works at the Moorfields Private Wing or whatever it is called as well as for the Moorfields NHS hospital.
I don't know the answers to all your questions, but that's one of them.
(Dentistry and opticians are private, because Thatcher took them out of the NHS, though a few dentists have a few NHS patients; if you are on benefits there are various concessions and free things, but again, I don't know about those because I am not on benefits. It still costs a fair bit to have bad teeth and eyes, she said bitterly.)
If I break my leg tomorrow, I can call an ambulance, be taken to the nearest appropriate hospital, and be treated, free. Ditto a heart attack or any other emergency treatment. Non-emergency takes longer: if my thyroid packs up, I see the doctor free but may have to wait a few days or even a couple of weeks for an appointment, and although treatment costs me nothing I have to see specialists at the hospital and that can take many months because they are horrifically overworked. I do have to pay a prescription charge for medication from the pharmacy, until I am over retirement age or if I were a child, or pregnant, or had one of about ten conditions which mean your prescriptions are exempted from payment.
I think the NHS is still the largest employer in Europe. It certainly used to be. And it is known that it could use 110% of the total national income and still have things it needed more money for.