No, since it's not an acronym. Opinions might differ on the period, but I'd say that since it's a spliced abbreviation (from Italian "onza"), no period is necessary. Then again, I don't put a period after Mr or Mrs either.

i would say do what looks right. rules like that are impossible to follow in all usages.

if it reads with absolute clarity and stylistic integrity, it's right.

I would say lowercase, not smallcaps, and definitely with a period after it: oz.

I recommend Fowler's rule that abbreviations should always have a period unless they end in the same letter with which the full word ends. Ergo:

12 onza -> 12 oz.

but

Saint John -> St John

Thank you so much!