Most instant or sorcery spells deal damage themselves, but a few cause other sources to deal damage. For example, a sorcery spell that says "Target creature you control deals damage equal to its power to any target" wouldn't cause Bloodfeather Phoenix's ability to trigger even if an opponent or battle was dealt damage, as it's the creature dealing the damage, not the spell.

You choose whether to pay as the triggered ability is resolving. Once the ability starts resolving, it's too late for anyone to respond by removing Bloodfeather Phoenix from your graveyard to stop you from bringing it back.

Bloodfeather Phoenix's last ability triggers only if Bloodfeather Phoenix is in your graveyard at the moment the instant or sorcery spell you control deals damage to an opponent or battle. For example, if you cast a sorcery spell that deals damage to an opponent and also enough damage to Bloodfeather Phoenix to destroy it at the same time, Bloodfeather Phoenix's ability won't trigger.