If the spell that's copied has an X whose value was determined as it was cast, the copy has the same value of X.

You can't choose to pay any additional costs for a copied spell. However, effects based on any additional costs that were paid for the original spell are copied as though those same costs were paid for the copy too.

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

Azula's ability and the copy it creates both resolve before the spell that caused the ability to trigger. They resolve even if the spell is countered before the copy is created.

Any choices made when the spell resolves won't have been made yet when it's copied. Any such choices will be made separately when the copy resolves.

The copy will have the same targets as the spell it's copying unless you choose new ones. You may change any number of the targets, including all of them or none of them. If, for one of the targets, you can't choose a new legal target, then it remains unchanged (even if the current target is illegal).

A resolving copy of a permanent spell becomes a token. That token isn't "created" and won't interact with abilities that care about tokens being created.