How will Quaithe reappear in the story? More importantly, will Quaithe reappear in the story? I envision a deus ex machina appearance at Vaes Dothrak before permanently disappearing into the shadows.

I think Quaithe will show up once in Vaes Dothrak to confirm some prophecies and point Dany to Westeros, and then I’m going to say once more at King’s Landing after Dany roasts Aegon, and then maybe once more during the Second Battle for the Dawn. 

But she wouldn’t be a deus ex machina – in Greek drama, these show up generally at the end of the play to resolve the plot. For example, in Euripedes’ Medea, they would lower a golden sun-chariot on a crane to help Medea flee the wrath of Jason having murdered his new bride, father-in-law, and his and Medea’s two children. And the chorus comes in to underline how the deus ex machina is about really blatant stuff happening unexpectedly:

     Zeus on Olympus,
     dispenses many things.    
     Gods often contradict
     our fondest expectations.
     What we anticipate
     does not come to pass.
     What we don’t expect
     some god finds a way                                                           
     to make it happen.
     So with this story.

Whereas Quaithe is really closest to the Greek chorus, warning the hero about the inexorable workings of fate and the perils of hubris, and then confirming that the will of the gods has come to pass in an unexpected ironic fashion. (She’s closest actually to the Oracle at Delphi, but the Oracle didn’t show up in Greek theater, so /shrug).