Wondrous Crucible

If you cast a spell “without paying its mana cost,” you can’t pay any alternative costs. You can, however, pay additional costs, such as kicker costs. If the card has any mandatory additional costs, you must pay those.

The card remains exiled no matter what happens to the copy.

If you don’t cast the copy (perhaps because there are no legal targets available or you don’t want to), the copy will cease to exist.