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

Due to a 2021 rules change to cascade, not only do you stop exiling cards if you exile a nonland card with lesser mana value than the spell with cascade, but the resulting spell you cast must also have lesser mana value. Previously, in cases where a card's mana value differed from the resulting spell, such as with some modal double-faced cards or cards with an Adventure, you could cast a spell with a higher mana value than the exiled card.

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

You exile the cards face up. All players will be able to see them.

A spell's mana value is determined only by its mana cost. Ignore any alternative costs, additional costs, cost increases, or cost reductions.

If there are any additional costs or cost increases that would apply to Ethersworn Sphinx, apply those before applying any cost reductions.

If a spell with cascade is countered, the cascade ability will still resolve normally.

Affinity can reduce only the generic mana in a spell's total cost. It can't reduce colored mana requirements.

The cost reduction is set before you pay any of the spell's costs. Specifically, you could lock in a discount for an artifact you control and then sacrifice that artifact to activate a mana ability.

The mana value of a split card is determined by the combined mana cost of its two halves. If cascade allows you to cast a split card, you may cast either half but not both halves.

Cascade triggers when you cast the spell, meaning that it resolves before that spell. If you end up casting the exiled card, it will go on the stack above the spell with cascade.

When the cascade ability resolves, you must exile cards. The only optional part of the ability is whether or not you cast the last card exiled.