You can’t activate “Dragged Down into the Grave” if either you, or your opponent, has no cards in hard other than “Dragged Down into the Grave”. But you can activate it if a player has no cards left in the Deck.
The turn player selects a card first for “Dragged Down into the Grave”.
You can activate “Appropriate” in response to “Dragged Down into the Grave” because drawing was the last thing performed.