Avioactive.
Good questions indeed and the very questions that have been asked of a Lisbon based fractional operator known for operating their aircraft 'privately' despite multiple ownership and a commercial AOC. I have seen them arrive empty using the aircraft registration as their callsign then leave with passengers under their company callsign.
Funnily enough when such flights are questioned by crew the flights are transferred to someone more compliant and the first crew never asked to do them again.
If you wanted to extend this conversation you could ask about things like fire cover too but maybe you answered the question when you mentioned the country in which they operate.