The first part of “Giant Kozaky” is a Continuous Effect that does not start a chain. The second part is a Trigger Effect that can be chained to.
You can chain “Barrel Behind the Door” to the Trigger Effect (the damage) of “Giant Kozaky”.
If “Giant Kozaky” is destroyed while face-down, then its effect that inflicts damage is not activated. If “Giant Kozaky” is face-up on the field and is targeted by “Tribute to the Doomed”, and you chain “Book of Moon” so that “Giant Kozaky” is face-down when destroyed by “Tribute to the Doomed”, there is no damage.
Like “Atomic Firefly”, “Foxfire”, and “Poison Draw Frog”, if a face-down “Giant Kozaky” is attacked and destroyed, its effect is not applied. However, if a face-down “Giant Kozaky” is attacked, and is not destroyed by the attack, if “Kozaky” is not face-up on the field, then “Giant Kozaky” is destroyed by its own effect after damage calculation, and damage is inflicted by its effect.
If “Drillroid” attacks a face-down “Giant Kozaky” while “Kozaky” is not on the field, “Giant Kozaky” is destroyed by the effect of “Drillroid” after being flipped face-up. The effect of “Giant Kozaky” is not applied and it does not inflict damage.
The effect of “Giant Kozaky” that inflicts damage to the controller activates in the Graveyard or the “removed zone”, wherever “Giant Kozaky” goes to after being destroyed. “The End of Anubis” will negate the effect of “Giant Kozaky” if it was destroyed and sent to the Graveyard, but “Banisher of the Light” will not (because “Giant Kozaky” does not have to be “destroyed and sent to the Graveyard”). Nor will “Skill Drain” negate the damage if “Giant Kozaky” is destroyed.
If “Giant Kozaky” is face-up on the field with “Kozaky”, and “Kozaky” is equipped to “Relinquished”, “Giant Kozaky” is not destroyed because a card called “Kozaky” is on the field.
If “Giant Kozaky” is equipped to “Relinquished” as an Equip Spell Card, “Giant Kozaky” will not be destroyed by its effect because it’s not a monster (even if “Kozaky” is not on the field).
If “Giant Kozaky” is equipped to “Relinquished” and is destroyed by “Mystical Space Typhoon” or the effect of “Relinquished”, the effect of “Giant Kozaky” will inflict damage to the controller.