Lazav Ability on the Stack
The Lazav that entered after Essence Flux is a new Lazav

Any old abilities that may be on the stack only see that the Lazav that they were supposed to change is no longer on the field.

This is similar to flickering the target of a spell or an ability to 'dodge' the ability (such as using Essence Flux to flicker a creature that is targeted by Murder) in that both interactions are governed by rule 400.7


  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 9 exceptions to this rule, and none of them apply here.

If Lazav changes controllers with its ability on the stack, that ability will still resolve.

In the secondary question (assuming Lazav wasn't flickered), the ability that targets Clone in your graveyard will resolve. Although Lazav changed controllers, the ability on the stack did not change controllers. "Your" in Lazav's ability on the stack still refers to you (the player who previously had control of Lazav), and Clone is still in your graveyard, so it is still a legal target. That ability will still find Lazav, even though it changed controllers, because it didn't change zones.