Great wall of text! o_o
I'm just going to confine this to health care for now.
How is health care defined?
Let's look it up. Efforts made to maintain or restore physical, mental, or emotional well-being especially by trained and licensed professionals —usually hyphenated when used attributively.
Must minor injuries be treated, or only major conditions?
That's needlessly black-and-white, don't you think? The clearest line is drawn to exclude cosmetic procedures that don't have anything to do with one's actual health. After that what you want to do is prioritize based on need or urgency (whichever word you prefer) and that's a very tricky thing to do. You obviously want the person with a life-threatening electrical burn to receive quick treatment over someone with a bump on their head. Think of it as a hierarchy of urgency.
What happens if there is one person who is a doctor? If every person is entitled to health care, then this doctor must provide that care. This doctor might be busy 24 hours a day. This might seem like a far-fetched situation, but logic must apply in every case.
Well, you partially answered your own question later on...
Perhaps we encourage people to become doctors by paying them more. This means higher taxes. But following logic, what happens if we still don’t have enough doctors? What happens if a doctor has been over worked and he decides he wants to take a day off?
You got the part of higher wages right but you completely missed the mark assuming that doctors can't be imported. Immigration of skilled labor is a welcome thing. We actually have that problem here as well and what ends up happening is that we see an increasing share of our doctors turning out to be immigrants. That said...
In order to satisfy our assumption, it would be logically valid to force the doctor, at gun point if need be, to help someone. Explain to me how this would not be permissible? Logic demands that it be so.
Seriously, this is just comical
I'll repeat myself for good measure. What ends up happening in practice is that when you have a shortage of doctors, more happens than just salaries going up. Doctors from abroad would see a lucrative opportunity and immigrate, getting that piece of the pie and filling the void. Sure, it's no step-function but it happens quickly enough for people to endure the shortage as correction is taking place. Immigration of skilled labor is a welcome thing. What's more, as the local population is seeing how lucrative a job doctors have, people are going to want those jobs. Sure, most adults might not be able to get educated into becoming a doctor but you can bet parents are going to want their children to consider such a lucrative career!
If your base principle says that you are entitled to X, then there must be a way for X to be provided. You cannot simply say, “A tax will go to pay the medical bills.” This assumes 1) that enough taxes will be available, and 2) enough doctors will exist. What happens if 1 or 2 are false?
Seeings as I already addressed 2), I'll just confine my answer to 1).
Remember what I mentioned earlier about prioritizing based on need and urgency? When funds are coming short, that's usually where the policy gets affected if tax revenues can't keep up and assuming that reallocation of tax funds is off the table. Funding for less urgent care such as pain killers get their funding slashed (that happened here), generic pharmaceuticals are given preference and "big name" pharmaceutical brands lose coverage (also happened here) and the elderly might also get their healthcare coverage removed from them (also happened here).
Seriously, there are more options available for solving such problems without turning to the Gulag. Sheesh.