No, the premiums are not subsidized. No, not everyone over 65 is covered. Every US citizen and legal resident is covered if they met the requirements. Rather than reiterate everything, I am giving you the link to the official Medicare web site for the qualifications. It is interactive.
Medicare.gov - Medicare Eligibility Tool (Eligibility Questions)
By the way, it is not automatic. You do have to apply for it.
The Medicare Advantage plans are offered by private insurance companies and must be approved by Medicare and follow their rules. They are not offered in all localities. Basically the government pays them $1000+ /month for each subscriber that qualifies for free Part A Medicare and is enrolled in part B. Medicare part B is optional and costs $96 /mo for everybody. The premium is higher for high income people ( $170,000 /yr for a couple ). The insurance company can offer whatever additions they want and can add additional premiums if they want. You can also check them out on the official Medicare web site. Many do not charge any premium. Whether they charge a premium or not is not an indicator of the quality. Most of the highest rated ones do not charge any premium.
Personally, I have had one of Health Net's Medicare advantage plans for 4 years. I pay no premiums, zero co-pays for any doctor ( general or specialist ), zero for labs or diagnostics etc. I do not pay any premium for prescription drug coverage. It even includes free gym membership. Each plan has a network of doctors, hospitals, etc. that you go to within your locality. The size of their network is an important factor to consider. My plan has 5000 doctors in its network. Other plans may only have 500. You do have coverage worldwide for any illness or injury requiring treatment.
Doctors, labs, etc. have a free choice whether they wish to belong to an insurance plan's network or Medicare. Most of them do.
If you don't meet qualifications for free Medicare part A then you can still enroll in it or a Medicare Advantage plan by paying a premium for it. This premium is not subsidized and won't be cheap. Most people that have regular Medicare and not an Advantage plan also buy a Medi-gap policy from an insurance company. It basically fills in the gaps not covered by regular Medicare and has a low premium.
Some of the medical insurance offered is from non-profit organizations. Also some states have their own health insurance.
I have only given you the basics. It by no means is a complete explanation. You can do your own research on the Internet by searching for "Medicare", "Health Insurance", or some other related term.