Is ‘ask them to both be there’ or ‘they both will be there’ ever grammatical?

Question

I am trying to articulate how to position the determiner/predeterminer ‘both’ behind the nouns being modified. Every rule that I came across on a cursory search involves some unspecified exception, so I have established this working guideline:

• if the nouns being modified are not within a complement clause and the sentence's first auxiliary verb (if any) is a single word, then insert ‘both’ right after this auxiliary verb;
• otherwise, insert ‘both’ immediately following the nouns being modified.

 Set A: it is always correct to put ‘both’ before its nouns
1a) Both x and y must be positive.
2a) Which of them is green? Both of them are green.
3a) Both her sons visited the dentist.
4a) Both patients have gingivitis.
5a) I asked both of you to stop doing that!
6a) It's time for both my dogs to be fed.
7a) Let both the birds be free.
8a) The fact is that both of them are culpable.
9a) Make both Jack and Jill sprint.
10a) Invite both of us!

 Set B: putting ‘both’ behind its nouns, based on the above guideline
1b) x and y must both be positive.
2b) Which of them is green? They are both green.
3b) Her sons both visited the dentist.
4b) The patients both have gingivitis.
5b) I asked you both to stop doing that!
6b) It's time for my dogs both to be fed.
7b) Let the birds both be free.
8b) The fact is that they both are culpable.
9b) Make Jack and Jill both sprint.
10b) Invite us both!

 Set C: putting ‘both’ behind its nouns, ignoring the above guideline
1c) x and y both must be positive. *
1d) x and y must be both positive.
2c) Which of them is green? They both are green.
5c) I asked you to both stop doing that! *
6c) It's time for my dogs to both be fed. *
6d) It's time for my dogs to be both fed.
7c) Let the birds be both free.
8c) The fact is that they are both culpable. *

I'm pretty sure that Set B is grammatically sound. Incidentally, without any italicisation or vocal emphasis, is Set B slightly more emphatic of its nouns' twoness/togetherness than Set A or do they have the exact same meaning?

As for Set C, however, every dictionary and style/usage guide that I skimmed through finds at least one—sometimes every—sentence in it to be grammatically nonstandard. Is any sentence in Set C—in particular, the four asterisked sentences—perfectly natural?

Answer 1

1. Both is just the dual allomorph of all, and can go wherever all can go (give or take the occasional of).

2. All (and therefore both) can undergo Quantifier-Float, moving from the noun phrase they bind to an adverbial position immediately after an auxiliary verb in a clause. In a series of auxiliaries like (is being), after the first auxiliary verb is the preferred position, and more common, but before the first, or after the second are not uncommon and both are perfectly grammatical.

3. As to the revised question: Sorry, your guideline was wrong; but that's science for you. So, yes, both questions in the title are already grammatical, though I can't say about for ever. I.e, there's nothing wrong grammatically either with

Ask them to both be there,
or with
They both will be there.