Who qualifies for an autopsy?

Depends: In most countries, any death that may be a homicide and/or reveal a danger to public health gets autopsied, and the medical examiner has discretion to autopsy any likely suicide, accident or body without a physician to certify the cause. Families can pay for one on any decedent; today a few hospitals still offer an autopsy to families as a courtesy and i wish this were more common.

Depends by state: Deaths in ohio are required to be reported to the coroner/medical examiner when "any person dies as a result of criminal or other violent means, or by casualty, or by suicide, or suddenly when in apparent health, or in any suspicious or unusual manner..." they will then investigate the circumstances and evaluate if an autopsy is necessary. However, family can always request one by hospital.