I have a face-down morph creature that has received 1 point of damage. If my opponent casts Shock or another thing to kill it, can I flip at that point to make the damage disappear or my card still have it (now flipped) until the end of my turn?
After turning the creature face up, the damage is still marked on the creature, it doesn't somehow disappear. It's still the same creature, only in a different 'state'.
120.6. Damage marked on a creature remains until the cleanup step
707.8. As a face-down permanent is turned face up, its copiable values revert to its normal copiable values. Any effects that have been applied to the face-down permanent still apply to the face-up permanent.
But if the creature has a high enough toughness (4 or more) or protection from red, it might still survive the Shock.
Note that turning a creature face up doesn't use the stack, so an opponent cannot cast a Shock in response to turning a creature face up to kill it while it's still morphed.