If the family prefers, a private pathologist can do the autopsy in the funeral home. It does not matter much whether the body has been embalmed first. Whoever does the autopsy, there should not be a problem with an open-casket funeral afterwards. This is true even if the brain has been removed and the dead person is bald. The pillow will conceal the marks.
Most religions allow autopsy.
If the body is that of an orthodox Jew, pathologists are happy to have a rabbi present to offer suggestions. Many Moslems prefer not to autopsy. If you have a question, check with your own religious advisor. Here's how an autopsy is done. In this example, there are three pathologists working together.