In American English we'd probably use the idiom "I'm not going to leave anytime soon" instead of "I'm not going to leave soon" which for some reason seems possibly awkward...I can't find any Google hits of anyone using it thus.
I am not going to go soon - has exactly the same meaning especially in British English.
In this lesson can Δεν πρόκειται να φύγω σύντομα also be translated as I'm not about to leave soon - having read the notes first I thought this would be an acceptable translation too?
Yes, indeed that would be a possible translation. It's been added, thank you ^.^
So...I am confused on which forms of verbs are used to form this structure with πρόκειται...φεύγω is "leave," right? Why is it φύγω in this phrase?