Does Etali keep exiled with GP?

Yes. Actually, as soon as Etali is exiled, The Akroan War’s impact ends; “management” solely impacts permanents or issues on the stack:

108.4. A card doesn’t have a controller except that card represents a everlasting or spell; in these circumstances, its controller is decided by the foundations for permanents or spells. See guidelines 110.2 and 112.2.

This additionally level to a solution to

below whose management does it return to if GP goes away?

which is “your opponent”:

110.2. A everlasting’s proprietor is similar because the proprietor of the cardboard that represents it (except it’s a token; see rule 111.2). A everlasting’s controller is, by default, the participant below whose management it entered the battlefield. Every everlasting has a controller.