Can an Epic Monster, i.e. the Nug with a Toughness of 5, be defeated remotely via Research Card (#6) where "1 Monster of your choice loses 2 Health", & the Tokyo section of an Asia Encounter Card (#4 & 5) where "1 Monster of your choice on any space loses 2 (or 3) Health"? Defeating the Nug solves one of the Shug-Niggurath Mystery Cards. Or, do Epic Monsters always have to be slain via direct Combat Encounter with an Investigator?

Yes. "Losing health" is a type of effect that is able to target Epic monsters. Encounters or effects that Epic monsters are not target table by are those referring to "didcard". You can cross-check this in the Ref guide, page 6

Can I insta-gib epic monsters with Diana? Given there is a cultist chillin' with the big guy, of course.

Rules state that you cannot discard an epic monster, so the answer is no.