If I have a sacrifice captain and a doctor on the same ship, can I sacrifice another crew, then save him with the doctor (with a net result of no extra action and the sacrificed crew being turned face down)?
No, because the Sacrifice ability only resolved if the crew sacrificed is removed from the game. This is because it is a cost to activate said ability. The “Doctor” ability only works on crew that would be eliminated.
Davy is correct. Both possible results of that combination are blocked. You can’t use them together to get unlimited free bonus actions, nor can you use them simply to flip crew facedown.