I am not a rules expert, but I would say that "no", the second question is not true.
Consider Rule 17.4:
17-4. Ball Resting Against Flagstick
When a player's ball rests against the flagstick in the hole and the ball is not holed, the player or another person authorized by him may move or remove the flagstick, and if the ball falls into the hole, the player is deemed to have holed out with his last stroke; otherwise, the ball, if moved, must be placed on the lip of the hole, without penalty.
That clearly only applies if the ball is not holed. See the definition of "holed":
A ball is "holed" when it is at rest within the circumference of the hole and all of it is below the level of the lip of the hole.
Although I can't find explicit verb-age stating that the holing of the ball concludes any play of the hole (seems obvious), many other rules that wave any penalties for a player for conduct after the ball has been holed. See Decisions on Rule 1-1
for a couple of them. It would seem strongly implied that when the ball is holed, it might as well vanish into thin air instantly because the hole is completed.
My conclusion is that once the ball is in the hole and at rest, the hole is done. Nothing else can apply toward that hole, except the addition of penalties for actions already taken.
Note that the ball must be at rest to be holed, if it's still jittering around while the flagstick is yanked and it pops out, my guess would be that it is still in play.