1)AFAIK, health insurance is only required if applying for a Schengen Visa, which vast majority don't need.
2)it's a requirement to get visa, thus airline doesn't need proof as the visa wouldn't be issued otherwise.
3)Again, the airline must ensure everything that you do to enter complies with US sanctions. They also must ensure the insurance meets Cuban requirements. None of this can be done with a privately bought policy.