Christmas-ish carols

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Across
  1. 3. Who laid an egg?
  2. 6. Joy to the world, but who is dead?
  3. 7. What did the Batmobile lose on the motorway?
  4. 8. Who smells?
Down
  1. 1. We three kings of Orient are, one in taxi, one in a car, one on a ... blowing his hooter
  2. 2. What item of clothing is the snowman flying without?
  3. 4. What channel were the Angels watching?
  4. 5. Or maybe the Angels were watching this channel?