Depends. Some are employed by the hospital, some are under contract with the hospital. Usually specialists and surgeons are under contract due to the high skill level and broad need, they will often have their own practice and contract with different hospitals or medical agencies. At least here in the US, this is why your billing following a procedure will come from the hospital and the different practitioners.

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Typically surgeons, proceduralists, and anesthesia are outside contractors. This varies and some or all can be hospital employees, especially when the hospital is an academic medical center.

This is how a surgeon can have what’s called “privileges” at multiple facilities.

Nope! Doctors are independent contractors. They also have a right called Refusal to Treat, that nurses don’t have the privileges of either.