If you pay the casualty cost of a spell, the copy will resolve before the original spell.

You may sacrifice only one creature to pay a spell's casualty cost, and you copy the spell only once.

The copy of the spell is created on the stack, so it's not “cast.” Abilities that trigger when a player casts a spell won't trigger.

Casualty N means “As an additional cost to cast this spell, you may sacrifice a creature with power N or greater.” and “When you cast this spell, if a casualty cost was paid for it, copy it. If the spell has any targets, you may choose new targets for the copy.”

If you cast Cut Your Losses and copy it targeting the same player, that player will mill half their library, then mill half of what remains after that. They do not mill their entire library.