When do copied abilities on a creature go away?
It will keep its abilities as long as it doesn't leave the battlefield. When it leaves the battlefiel, it is considered a new object:


  400.7. An object that moves from one zone to another becomes a new object with no memory of, or relation to, its previous existence.
  There are seven exceptions to this rule:
  
  400.7a Effects from spells, activated abilities, and triggered abilities that change the characteristics of a permanent spell on the
  stack continue to apply to the permanent that spell becomes.
  
  400.7b Prevention effects that apply to damage from a permanent spell on the stack continue to apply to damage from the permanent that spell
  becomes.
  
  400.7c If an ability of a permanent requires information about choices made as that permanent was cast as a spell, including what mana was
  spent to cast that spell, it uses information about the spell that
  became that permanent as it resolved.
  
  400.7d Abilities that trigger when an object moves from one zone to another (for example, “When Rancor is put into a graveyard from the
  battlefield”) can find the new object that it became in the zone it
  moved to when the ability triggered, if that zone is a public zone.
  
  400.7e Abilities of Auras that trigger when the enchanted permanent leaves the battlefield can find the new object that Aura became in its
  owner’s graveyard if it was put into that graveyard at the same time
  the enchanted permanent left the battlefield. It can also find the new
  object that Aura became in its owner’s graveyard as a result of being
  put there as a state-based action for not being attached to a
  permanent. (See rule 704.5n.)
  
  400.7f If an effect grants a nonland card an ability that allows it to be cast, that ability will continue to apply to the new object that
  card became after it moved to the stack as a result of being cast this
  way.
  
  400.7g A resolving spell or activated ability can perform actions on an object that moved from one zone to another while that spell was
  being cast or that ability was being activated, if that object moved
  to a public zone.


Therefore, in the graveyard, it won't be a copy anymore, so you can't scavenge it.

Leaves-the-battlefield triggered abilities will work, because the game looks for the characteristics of the object on the battlefield. For example, persist will work.


  603.6c Leaves-the-battlefield abilities trigger when a permanent moves from the battlefield to another zone, or when a phased-in permanent
  leaves the game because its owner leaves the game. These are written
  as, but aren’t limited to, “When [this object] leaves the battlefield,
  . . .” or “Whenever [something] is put into a graveyard from the
  battlefield, . . . .” An ability that attempts to do something to the
  card that left the battlefield checks for it only in the first zone
  that it went to. An ability that triggers when a card is put into a
  certain zone “from anywhere” is never treated as a
  leaves-the-battlefield ability, even if an object is put into that
  zone from the battlefield.
  
  603.6d 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, abilities that trigger when a
  permanent phases out, abilities that trigger when an object that all
  players can see is put into a hand or library, abilities that trigger
  specifically when an object becomes unattached, abilities that trigger
  when a player loses control of an object, and abilities that trigger
  when a player planeswalks away from a plane 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.


However, remember that abilities like the one in Emrakul, the Aeons Torn are not Leaves-the-battlefield abilities, because they tigger when the permanent is put on the graveyard from anywhere, not only from the battlefield. Therefore, that ability won't work if you copy Emrakul.