You pay all costs and follow all normal timing rules for a card played this way. For example, if the exiled card is a land card, you may play it only during your main phase while the stack is empty.

Fires of Mount Doom's first ability can target a creature that has no Equipment attached to it.

If you activate Fires of Mount Doom's last ability but don't play the exiled card this turn using that casting permission, Fires of Mount Doom won't deal damage to each player.

If the target of Fires of Mount Doom's first ability is illegal by the time the ability tries to resolve, it's removed from the stack. You won't destroy any Equipment.