1a. Do you trust them?
1b. Don’t you trust them?
As @SimonB has pointed out, only (1b) is an actual "Negative Question"; I'll return to that below. The other examples are also interesting, and cause a lot of trouble, so they're worth considering, too.
2a. Has anyone told you what to say?
2b. Has someone told you what to say?
(2a-b) illustrates the some/any distinction; logically, some means the same thing as any (both are the existential quantifier ∃), just like, logically, but means the same thing as and (both are the dyadic functor ⋀). But any is a Negative Polarity Item -- perhaps the NPI, since it's often used as an example -- and it can only appear grammatically in a negative context.
That can mean after a negative, like
- She didn't mean anything by that.
whereas without a negative the sentence would be ungrammatical,
- *She meant anything by that.
but an equivalent with some would be OK.
- She meant something by that.
So why is it OK in a question, then? Well, yes/no questions are negative contexts -- they alway have the possibility of reference to negation. That means you can get questions with either some or any. Some people use that distinction, along with intonation and stress, to indicate their own presuppositions.
For instance, I would consider (2b) as being conversationally (depending on how it's pronounced, of course) equivalent to
- Is there someone who told you what to say?
with an invited implication that there might well be someone
and an additional implication that you would be asked about them, too
as opposed to (2a), which is neutral. Again, negation, in the form of NPIs and their usefulness in other occupations, is important. But they aren't negative questions, since no overt negation is involved; only NPIs.
3a. She is quite clever.
3b. She is quite clever, isn’t she?
(3b) has a negative in it, but it's not a negative question. It's what's called a Tag Question, which can be formed from almost any declarative sentence by repeating the first auxiliary verb (using Do-support as needed) in the negative if it was originally affirmative, and the affirmative if it was originally negative. It also requires a pronoun for the subject, repeated right after the auxiliary.
- It's midnight, isn't it?
- It isn't midnight, is it?
In each case, the statement shows the speaker's belief (or hope) and the question (which can use a variety of intonations for a variety of effects) asks for some kind of agreement, or clarification, or attention. or ... from the addressee. Again, not exactly negative questions, though negation is part of the structure.
Now, back to negative questions. Since a yes/no question always has the possibility of negation, what's the purpose of allowing negatives in them at all? Good question. The answer is that there's no way to forbid it -- the rules are usable -- and there's a use for it. Like the use of some instead of any in a question, even though there's no negative, the use of Don't instead of Do in (1b) changes the perceived presupposition of the questioner. The speaker of (1b) is at least inviting the inference that you don't trust them, and angling for confirmation. (1a), without the unnecessary negative, is once again neutral.
tl;dr Negation is important.
(If any of the terms I use are strange to you, you can search for them in the box at the top of the page)