"Until your next turn" means the effect wears off as soon as your next turn starts, even before you untap. It's basically the same as "until the end of the turn of the player who immediately precedes you."

The card is cast using all normal rules and restrictions for casting spells. Casting it means you'll have to pay its mana cost, and it will go to the graveyard after it resolves or is countered. The only thing that's different is you're casting it from the Exile zone.

If you don't cast the card, it stays exiled.