The last ability of Izzet Chemister only lets you cast cards exiled by that card. If Izzet Chemister leaves the battlefield before you activate that ability, the exiled cards are lost forever.

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, such as that of Tormenting Voice, those must be paid to cast the card.

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

You must cast any of the exiled cards you wish to cast while Izzet Chemister's last ability is resolving. You can't cast them later in the turn. You can cast a sorcery this way at a time you normally wouldn't be able to.

You cast the exiled cards one at a time in any order. Later spells can target ones you cast earlier. None of the spells resolve until after you're done casting them all.

Any of the cards you don't cast will remain in exile.