Question about The Shallow Grave (Spell Card)
Based on this ruling on Disappear:


  When a card that selects a card in the Graveyard as a target, such as "Monster Reborn", "Mask of Darkness", "Magician of Faith", "Spear Cretin", or "The Shallow Grave", is activated, you may chain "Disappear" to remove the targeted card from play, and the "Monster Reborn", etc., will resolve without effect. This is because these cards select their target at activation, not resolution.


(Source)

Your opponent wouldn't get the monster.

D.D. Crow appears to work exactly like Disappear when discarded.  So the same ruling should apply.