Does turning a card face up cause it to lose any counters it had on it?
Any enchantments or counters that were on the creature when it was face down will remain when it is turned face up. The creature never changed zones. It is still the same object.

Morphs and Manifests have the "Creature" card type, but no subtypes.


  701.31a, 702.36a It becomes a 2/2 face-down creature card with no text, no name, no subtypes, and no mana cost.


When it gets turned face up, it just regains its normal characteristics, but it is otherwise the same object:


  702.36e Any time you have priority, you may turn [a morphed card] face up. The morph effect on it ends, and it regains its normal characteristics.


Transform means something special and is not what happens with morphed cards. I like to say "morph" and "turn it face up".