Speaking personally... if I ever had an NPC actually go through the effort of knocking out a PC, it's probably because of some story reason. Which means, no, I wouldn't let the PC try and wake up on their own... because the PC will wake up in the prison cell, or tied up in the hold of the boat, or wherever it was the NPC was trying to get the PC to by knocking him out in the first place. They're being knocked out so that they can wake up some place specific. A randomly-determined premature waking up serves no purpose.
No, because you open a can of worms with the Death Saving Throw in general. By the same logic, you should allow someone who has already made their 3 Saves (thus Stable) to continue making checks to see if they wake up. This makes the rest of the rules on being Stable irrelevant, since you should be able to wake up and be ready (with 1 HP) in about 2 minutes or so.IF an NPC tries to knock out a PC (or vice versa) and reduces her to 0 hps she is unconscious and stable. Would you allow her to make Stabilization rolls anyway in hopes she can roll a 20 and gain an HP and wake up?

(Dungeons & Dragons)
Rulebook featuring "high magic" options, including a host of new spells.