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

The creature that entered the battlefield from your graveyard deals damage equal to its current power (including any +1/+1 counters it entered the battlefield with) to the target creature 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.

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 the battlefield 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.

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.

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