Flayer of the Hatebound's last ability will trigger even if a creature enters from your graveyard under another player's control.

Flayer of the Hatebound is the source of the ability, but it may be another creature that is the source of the damage. If a black creature enters from your graveyard, the ability could target a creature with protection from black, although the damage will be prevented. It couldn't target a creature with protection from red.

If you cast a creature card from your graveyard, that card will be put on the stack before entering. Flayer of the Hatebound won't trigger.

Since damage is dealt by the creature, abilities like lifelink and deathtouch are taken into account, even if the creature has left the battlefield by the time it deals damage.

The creature that entered from your graveyard deals damage equal to its current power (including any +1/+1 counters it entered with) to the target permanent or player. If it's no longer on the battlefield when the ability resolves, its last known existence on the battlefield is checked to determine its power.