Hmm... in most cases with a "not" in place, I'd just use "or," e.g. "I do not like pickles or onions." BUT if I were balancing two complete negative clauses -- e.g. "I do not like pickles" and "I do not enjoy onions, either," I might very well combine them like this:
I do not like pickles, nor do I enjoy onions.
As far as the question of what you are allowed/required to do goes, people will disagree with each other, but the Oxford English Dictionary's page on "nor" suggests that you can basically always get away with using the "nor" -- even sometimes if there isn't a "not" -- and it just sort of comes down to the fact that you'll sound like a seventeenth-century poet lots of the time if you do so.
In my own writing, I stick mostly to using "or" unless I have either a "neither" justifying the presence of a "nor" or two complete negative clauses wanting to be linked with a "nor." But whether you use "nor"s in other situations is mostly a matter of personal preference, and you can find support in various sources for whatever you decide (and, doubtless, you can also find sources saying that whatever you've decided is emphatically wrong!).
I am VERY confident the GMAT will not force you to make a decision with respect to "not...nor," since not much can be definitively considered wrong with respect to that construction. Hypothetically, if it did come up, you'd employ the strategy that really applies to all SC questions: if there are two options both free of grammar errors, your last-resort focus should shift to style. If there were a sentence that sounded better (e.g. one that used the more standard "neither...nor" about which you had no doubts), you'd pick that sentence. If there were no better alternative, you'd pick the sentence with the questionable-sounding "not...nor."
Cheers,
Ashley Newman-Owens
GMAT Instructor
Veritas Prep
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