This one I can answer although I’m not a doctor - I’m related to a few. Most doctors aren’t employees, they’re independent contractors who sign on to “get privileges” to work at a hospital. Doctors aren’t paid by the hospital, they’re paid by the patients. A hospital assigns patients to doctors who are qualified and available to treat patients. Here’s an article.
Malpractice isn’t an insurance provided by any hospital doctors have privileges at. Doctors pay out of pocket, and it reduces their take home pay. Malpractice rates vary based on where a doctor lives and works, type of field, and amount of coverage. Many pay between $2,000 to $4,000 per year, but some may pay as much as $15,000 to $20,000 a year. Extremely risky areas may be even higher - with OB/GYN listed in this article at $40,000 - for a doctor with no claims against him. So, some high earning doctors make pay out a lot just to be able to practice safely.
Situations may arise that are hospital-related (like a building-wide infection), and in those cases the hospital will take care of the insurance and legal coverage.
So tl;dr - Doctors at hospitals and with their own practices pay for their own coverage.
