"His cat is playing with the dog."
Translation:Seine Katze spielt mit dem Hund.
Maybe it helps if you remember that accusative is related to the English word accuse. If you could replace the verb with acuse it si accusative. "He paints the door" could also work as "He accuses the door" (don't think about the sense) In "He plays with the dog" play cannot be replaced with accuse: "He accuses with the dog" is not a correct sentence