@ Bananasbananas, We essentially agree with each other. But of your three hypothetical questions, the first and second are irrelevant - if a person is fatigued, they are fatigued - no matter what the reason. Of the third; if your sleeping preparations were diligently applied - black-out blinds in the bedroom etc, there should be no come-back - from a decent airline. If your neighbours were having a barbecue or a barking dog or neighbour's music kept you awake, that is perfectly legitimate reason for reporting unfit in the first place.
I put in a number of fatigue reports at a previous airline over the years, all of which were properly investigated, but medical intervention was never needed or requested. I reported fatigue and my level of fatigue, according to a comprehensive list of fatigue indications and descriptions, and the fatigue management team took it from there and looked at my rosters etc.
During a turn-around, a check from a doctor or medic would not be possible quickly, it would incur an even greater delay that would most probably stretch beyond the allowable discretion period for the rest of the aircrew. It is illegal to get airborne if you know beforehand that your flight will go beyond the allowable period of discretion.
Airlines know all this so they put pressure on the crew members to self declare, as per your three questions; (knowing that most won't bother), and also onto the Captain's shoulders, who is in a very difficult place, being responsible for the welfare of their crew but also the requirements of the airline and the potential financial consequences of a delayed flight.