Breaching Dragonstorm

If the spell you cast has {X} in its mana cost, you must choose 0 as the value of X when casting it without paying its mana cost.

If you choose to cast the exiled card, you do so while Breaching Dragonstorm’s first ability is resolving and still on the stack. You can’t wait to cast it later in the turn. Timing restrictions based on the card’s type are ignored.

Since you are using an alternative cost to cast the spell, you can’t pay any other alternative costs. You can, however, pay additional costs, such as kicker costs. If the card has any mandatory additional costs, you must pay those.