Is "afero" used as often as "thing" is in English? If the object(s) in question are ambiguous, like "Go get your things," or unknown, as in "What is that thing?" would "afero" be used?
Afero is not a physical thing. If you're talking about objects (e.g. go get your things), you'd use aĵo. Iru por preni viajn aĵojn.
Afero is more abstract. It means 'matter', if that makes sense. I know I'm not doing a great job of explaining it, but I hope that helps somewhat. :D
I searched several bilingual dictionaries and here is the one that gave the broadest range of definitions https://en.glosbe.com/eo/en/afero . the consensus is "affair, matter, thing, business,", but this one includes many other synonyms, including "case". Seems that "situation" is not a close enough synonym (although it is only one degree of separation away from "affair": http://dictionary.reference.com/browse/situation?s=t )
And yet I just got marked wrong for "It's not a serious matter" - is there a distinction I'm missing or should I report this as an answer that should have been taken for correct?
It's a different grammatical construction.
The matter is not serious = La afero ne estas grava.
subject = The matter/La afero
complement = serious/grava
It is not a serious matter = Ne estas grava afero.
subject = It/∅
complement = a serious matter/grava afero
Do you see how in one, the adjective is not part of the noun phrase and in the other it is part of the noun phrase?
I used the sentence "la afero estas malgrava" and that was accepted... I suppose "la afero malgravas" is acceptable as well.
Yes, but only for the very narrow definition of "case" that means the same as "matter" or "affair"