Right or wrong, the version I see most in print is "do's and don'ts". Perhaps this is because "dos" looks like Spanish for "two" or Microsoft's precursor to Windows.

The archaic phrasal verb "to do part" means "to separate".

I would classify "correspondence", in the sense above, as a non-countable noun. I think "correspondences" might be usable in some circumstances, but I can't think of any in which it would be preferable to "correspondence".

Certain words, like "equipment", have no plural form at all.

its an in-frigging-fix.

Yes, tmesis or infix.

B-ass-ackwards is another example. Unbe-fucking-lievable, isn't it?

"societal" is used by people who are afraid of the word "social" because it makes them think of socialism.

"Homely" means "ugly" in the US, "homey" in the UK. It's as simple as that.

In instructions, there is no question mark, as it's a statement:

"How to operate this machine" means "This is how to operate this machine" or "Here's who to operate this machine".

But in captions in forums and the like, I see it as highly informal yet acceptable shorthand. Elsewhere I would object to its use. It should never be used in "real" writing.

I've mostly seen objections to "impact" as a verb, with the caution that "affect" should be used in its place. Impact most decidedly is a noun, whose verbing is a relatively recent phenomenon.

I'd say Porsche has elaborated the distinctions that seem right to me. Societal is first listed in the OED for 1898 in fact. And you have the option of the even older world "societary" with a more or less synonymous meaning. Try that one out on your professor perhaps?!