Depths of Despair (Red)

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When this defends, banish it when the combat chain closes.

Blood Debt

The Disciple of Pain sought dominion, but in desperance he languished.

Rulings

From the Compendium of Rathe release notes

Depths of Despair’s first ability triggers when it becomes a defending card on the combat chain. When the triggered effect resolves, it generates a delayed triggered effect that triggers when the combat chain closes. You will banish it when that specific triggered effect resolves.
Neither player receives priority during the close step of combat.

Keyword notes

Blood Debt - Blood Debt is a new keyword existing on Shadow cards. Shadow cards can inflict a loss of life to access a greater power. At the end of your turn, you lose 1{h} for each blood debt card in your banished zone.

You only lose life at the end phase. Players do not have priority during the end phase.
If you have multiple cards with blood debt, you lose 1 for each instance of blood debt at the end of turn.
Only face-up cards banished with blood debt counts and will lose you . Face-down banished cards have no effects or stats.