If the target creature is an illegal target when Fiery Annihilation begins to resolve, the target Equipment (if any) won't be exiled.

Fiery Annihilation's replacement effect will exile the target creature if it would die this turn for any reason, not just due to lethal damage.

Fiery Annihilation can target a creature that has no Equipment attached to it.

If the target Equipment is no longer attached to the target creature when Fiery Annihilation resolves, it won't be exiled.