Can a family member get paid as a caretaker for their elderly father in Florida, and can this be paid out of the estate?

Until your father passes away there is no estate....but, I think you mean can your Dad pay you? Yes.

be sure you have a caregivers contract. Make sure it itemizes the hours of work, those thing you are responsible for....if transport is needed...besure you include the cost of that and that you are not ever financially responsible to provide alternate transport if you cannot do it.

make sure you get paid through regular payroll...social security, Medicare, taxes, etc. withheld.

at some point in time there is likely to be other in positions of authority that will want to see the bookkeeping for everything...so keep good records and receipts.