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"Flashback [cost]" means "You may cast this card from your graveyard by paying [cost] rather than paying its mana cost" and "If the flashback cost was paid, exile this card instead of putting it anywhere else any time it would leave the stack."

A spell cast using flashback will always be exiled afterward, whether it resolves, is countered, or leaves the stack in some other way.

If X is 0, Conflagrate has no targets and deals no damage.

If a card with flashback is put into your graveyard during your turn, you can cast it if it's legal to do so before any other player can take any actions.

If a spell has {X} in its mana cost, include the value chosen for that X when determining the mana value of that spell, even if it was cast with flashback and no mana was spent on X.

If some of the targets become illegal for Conflagrate, the original division of damage still applies, but the damage that would have been dealt to illegal targets isn't dealt at all.

To determine the total cost of a spell, start with the mana cost or alternative cost (such as a flashback cost) you're paying, add any cost increases, then apply any cost reductions. The mana value of the spell is determined only by its mana cost, no matter what the total cost to cast the spell was.

You can cast a spell using flashback even if it was somehow put into your graveyard without having been cast.

You divide the damage as you cast Conflagrate, not as it resolves. Each target must be assigned at least 1 damage. You can't choose more than X targets and assign 0 damage to a target.

You must still follow any timing restrictions and permissions, including those based on the card's type. For instance, you can cast a sorcery using flashback only when you could normally cast a sorcery.