What happens if I activate the ability with X=2 targeting Lazav in my graveyard?
Is its name Lazav, the Multifarious or Necrotic Ooze?
How many activated abilities would it have?
When Necrotic Ooze becomes a copy of Lazav with the activated ability Necrotic Ooze gained from Lazav, then Necrotic Ooze—
will have the name Lazav, the Multifarious (not any other name, including Necrotic Ooze),
will be legendary in addition to its other types, and
will have another instance of that ability, but
will no longer have "As long as Necrotic Ooze is on the battlefield, it has...", so that in general, it won't have any activated abilities of cards in graveyards anymore and the only activated abilities it would have are two instances of "X: Lazav, the Multifarious becomes..." (one from being a copy of the original card and one from the exception) (remember that a copy effect overwrites an object's copiable values, with only those exceptions given by the effect)
(C.R. 706.2, 706.9, 706.9a, 706.9c, 201.4a-b, 201.4, 205.1a). Where that activated ability says "Lazav, the Multifarious becomes...", read it as "[This permanent's name] becomes..."; however, where that ability says, "except its name is Lazav, the Multifarious", that remains unchanged (this instance of "Lazav, the Multifarious" doesn't refer to the object it's on by name, unlike the other instance) (C.R. 201.4b).
So my Ooze has all activated abilities of Lazav.
What happens if I activate the ability with X=2 targeting Lazav in my graveyard?
Is its name Lazav, the Multifarious or Necrotic Ooze?
How many activated abilities would it have?
See also this thread and this thread.
EDIT (Apr. 5): Correctness edit.