Does killing a face-down morph trigger the death triggers printed on the card?
Ashcloud Phoenix's ability does not trigger, and it stays in the graveyard.

Rule 603.6d says


  Normally, objects that exist immediately after an event are checked to see if the event matched any trigger conditions. Continuous effects that exist at that time are used to determine what the trigger conditions are and what the objects involved in the event look like. However, some triggered abilities must be treated specially. Leaves-the-battlefield abilities... will trigger based on their existence, and the appearance of objects, prior to the event rather than afterward. The game has to "look back in time" to determine if these abilities trigger.


Since the phoenix did not have the ability before it was destroyed, the ability does not trigger.