If I copy Etrata, the Silencer, but not its name as in the case of Lazav, the Multifarious; will its copied abilities work since Etrata's ability states "When Etrata deals damage" and Lazav doesn't copy Etrata's name? Or would it require the card to possess both the ability and name?
Its copied abilities would definitely work, even though Lazav is still named Lazav. See rule 201.4:
201.4. Text that refers to the object it’s on by name means just that particular object and not any other objects with that name, regardless of any name changes caused by game effects.
201.4b If an ability of an object refers to that object by name, and an object with a different name gains that ability, each instance of the first name in the gained ability that refers to the first object by name should be treated as the second name.