If you cast any of the exiled cards, you do so as part of the resolution of the triggered ability. You can’t wait to cast them later in the turn. Timing restrictions based on a card’s type are ignored.

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

If an exiled card 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 cast more than one of the exiled cards, you choose the order in which to cast them. A spell you cast this way can be the target of a later spell you cast this way. However, permanent spells cast this way won’t resolve until you’re done casting spells, so the permanents they become can’t be the target of spells cast this way. For example, if you exile Twincast and Lightning Strike, you can cast Lightning Strike and then cast Twincast targeting it; but if you exile a creature card and an Aura card, you can’t cast that Aura targeting that creature.

Any cards not cast, including land cards, remain in exile. They can’t be cast on later turns.

In a multiplayer game, if a player leaves the game, all cards that player owns leave as well. If you leave the game, any spells or permanents you control from Etali’s ability are exiled.

Losing the game because a player (preferably an opponent) has ten or more poison counters is a rule of the game. Etali, Primal Sickness doesn’t have to still be on the battlefield when someone (preferably an opponent) gets their tenth poison counter.