So, does this have any connection to Zazu from Lion King? A cognate from another Bantu language or perhaps a variation?
I don't think so. From what I could find Zazu is not of Swahili origin, but rather Hebrew. Not all characters had Swahili names, viz. Ed (one of the hyenas).
The translation for other M- nouns say that the "a" or "an" article is required. Are there language rules about this, or is it a quirk of the course?
Quirk of the course. a/an; the; or no article versions should be accepted for most straight noun phrases I should think. Report away. Hopefully this issue will be quickly resolved.
Is "mzazi" a term someone would casually use to refer to their own parent, or does it have the more formal/disconnected connotation that it does in english in swahili too?
Mzazi means parent, the biological one that is.