The Florida Constitution does have a specific clause guaranteeing a right to privacy. I'm not a legal expert and cannot cite the article or section, but I know that it does exist. The article is unclear (typical for the agenda-driven rubbish that passes for "journalism" these days; never let facts get in the way of spin) about whether this action is in state or federal court, but I'm pretty sure it's state.
Yes it does, and a specific statute governing medical privacy.
States can grant citizens rights in addition to federal rights.