I would have written Nem tanár vagyok. Why the “subject(hidden), verb, object” word order in this case?
Your sentence is a correct translation. The preferred one has the neutral word order, yours emphasizes "tanár" (or rather "nem tanár"). "I am not a teacher (but something else)"
All in all, "Nem tanár vagyok." should be acceptable.
Would 'Nem én tanár' because wouldn't being a teacher technically be a quality?
You only drop the verb in the third person ( van, vannak), not the first or second person.
So, I can either use the Pronoum "Én" or not, since the verb points to the person who's speaking, right? My point is "Én nem vagyok tanár" would be correct and mean the same thing.
Correct. If you would like to emphasize that you are something else (you are surprised that somebody thougth you are a teacher), then just put the "én" out. Én nem tanár vagyok, hanem diák - I'm not a teacher but a student.