If you cast a spell "without paying its mana cost", you can't pay any alternative costs. You can, however, pay additional costs. If the spell has any mandatory additional costs, those must be paid to cast the card.

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

If you cast an instant or sorcery spell, you do so as part of the resolution of Sorcerous Squall. You can't wait to cast one later in the turn. Timing restrictions based on a spell's type are ignored.

Delve doesn't change a spell's mana cost or mana value. For example, Sorcerous Squall's mana value is 9 even if you exiled three cards to cast it.

You can exile cards to pay only for generic mana, and you can't exile more cards than the generic mana requirement of a spell with delve. For example, you can't exile more than six cards from your graveyard to cast Sorcerous Squall unless an effect has increased its cost.

Because delve isn't an alternative cost, it can be used in conjunction with alternative costs, such as flashback. It can also be used to pay for additional costs that include generic mana.