If Mirage Mirror becomes a copy of a creature, and then is put into the library some how, does it still stay a creature until end of turn?
No, it's just an ordinary Mirage Mirror artifact once it's in your library. It won't still be a copy of anything.
The key rule here is:

400.7. An object that moves from one zone to another becomes a new object with no memory of, or relation to, its previous existence. There are nine exceptions to this rule:
(None of the exceptions apply. They're mostly about cards keeping track of each other between zones when they need to.)

Generally after something moves to a new zone, it's reset to a fresh state of just being whatever the card says.