You're quite right that normally the standard sequence never ever would be used (where ever is simply an optional intensifier for never). But for your second example, including ever actually alters the intended meaning.

1: Did you never need to use that pickup line?
In this version, ever would normally be understood to mean much the same as ever...
(Was there not even a single situation where you used/needed to use it?)

2: Did you ever never need to use that pickup line?
Here, the unusual sequence forces us to apply a different interpretation...
(Was there not even a single situation where you didn't [need to] use it?)

You're quite right that normally the standard sequence never ever would be used (where ever is simply an optional intensifier for never). But for your second example, including ever actually alters the intended meaning.

1: Did you never [ever] need to use that pickup line?
In this version, never would normally be understood to mean much the same as ever anyway...
(Was there not even a single situation where you used/needed to use it?)

2: Did you ever never need to use that pickup line?
Here, the unusual sequence forces us to apply a different interpretation...
(Was there not even a single situation where you didn't [need to] use it?)

Another way of looking at it is that in #1 the scope of never is all occasions when the addressee was trying to chat someone up. In #2, the scope is a single "chatup" approach.