After Flawless Forgery resolves, the exiled card remains exiled.

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

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.

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