I know that citizens from another countries, who want to

To  stay permanently ,  in   USA ,  must have with all their papers and medical exams . What medical exams must do and have with them , so they  can  live  in USA ?
Someone from Greece what medical exams must do and have with him  ?

First off. If you want to stay permanently in the USA. You will either need a green card(Permeant residence), or you will need a visa.

Some of the visas that allow you to stay are multi-entry tourist visas which can be for 1 year, 5 years, or 10 years. Not many people get these and these are usually for Europeans. Anyone from Africa, Asia, etc will most likely never get these, and if they do its rare.

If your a European or a country that can travel visa free to the USA then you could just keep crossing the border to Canada, or Mexico every 30-90 days or more depending on how long you can stay visa free in the USA.

If you have a H1B visa(3 years), or some other work visas then you can work in the USA. But they do not last forever, and they need to keep being renewed by your employer.

If you have money you can try for the investor visa where you invest 500 thousand USD or more into the USA, and it gives you an instant green card(Permant residence).

There is many other methods to. But depending on what country you are from is going to decide how long you can stay in the USA or not.

You will need to post more information on the people, and what countries they come from.

What about sponsorship?  Is that still a thing?  If so, how does it work?

What about sponsorship? Is that still a thing? If so, how does it work?
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Yes. Sponsorship is a thing. But the sponsorship means only that the person is finically sponsoring you if you don't have enough money to meet the requirements fora visa. They can only directly sponsor you if you are a family member of them or a spouse.

There now has been exceptions made for two nationalities according to immigration law where they don't need to be a family member or a spouse. Which is Ukranian or Venezuelan.