None of these uses are clear, idiomatic English.
What is the correct tag question of proverbs like (a) None is none under the sun,isn't it/ are they? (b) Time and tide wait for none, isn't it/ do they? (c) Waste not, want not, isn't it?
I've given the answer with confusion. I think that as the proverbs are universal, the tag question should be always 'isn't it'.
Isn't it would require some antecedent for "it" that does not exist in some of these examples. If you mean, "Isn't it the case that this is true or applicable?", you'd need to expound at least a little, saying isn't it so?. Equivalent, colloquial tag questions with the same basic meaning here are right?, eh?, huh? You could say, Waste not, want not, right?
If you wanted, you could construct a tag question that agrees with the actual content of the proverb. As you suspect, since the proverb is a set phrase, the generic tag question would be better. Still, you could, if you really wanted, say "None is none under the sun, isn't it?" and "Time and tide wait for no man, don't they?" Since waste not, want not is in a command form, I don't think there is any good way to add a tag question that agrees grammatically with the content.
Problem 2: What is the correct tag question of interrogative sentences like: (a) Who cares, do they?
"Who cares, do they?" sounds like wordplay. It isn't a suitable tag question.
Since Who cares? is a common rhetorical question, it makes sense to tag it generically, as in, "Who cares, right?"
If it were a real interrogative question, no substantive tag would apply. You would not say, "What is her number, right?" nor "What is her number, is it?" These are not grammatical.
In colloquial speech, some speakers might add a hunh, eh, hmm, or similar at the end of some interrogative questions. I don't recommend trying to adopt this practice.
Problem 3:Can we add tag question to phrases like (a) Good morning, ......? (b) Happy birthday, ....?
Not as such. I don't understand what this would do.
With good morning, you could certainly ask the question, "Good morning, isn't it?" or "Good morning, wouldn't you say?" or "Good morning, I hope?" Though these start with a clause good morning, I would not consider that clause to be equivalent to the greeting good morning, per se.
Happy birthday is less suitable to such twisting around.