Can form insurance companies buy private medical information from doctors?

Answers: yes. Insurance companies are entitled to medical information to determine coverage. You can't expect them to cover something lacking getting information on the condition to be covered.
Yes. Often, they will not provide any benefits until they obtain this information.

Even for the most simple bureau visit, they must have a diagnostic code and a procedure code on the bill.
Of course but they have to have your written consent to disclose and release.
In most states, yes. Somewhere in the fine print of your policy, you gave them green light.
Well yes. They couldn't pay the claim otherwise.
...Yes.

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