Your alternative "Her maiden name is not Atkins either Perkins" is definitely wrong. The original is weird, and likely incorrect although I guess it might function in certain obscure contexts:
"I am completely sure her maiden name is either Atkins or Perkins."
"Unfortunately, that's entirely wrong yet again. Her maiden name is not 'either Atkins or Perkins.' It's Xian-Do, an esteemed Chinese family."
Normally, you use constructs
- either X or y for positive choices (one of two)
- neither X nor y for negative choices (both are wrong).
In your case the sequence should be:
Her maiden name is neither Atkins nor Perkins
You can also use 'neither' and 'either' to refer back to earlier choices:
I think her name was Atkins or Perkins, but if I were to decide, I must say either sounds likely.
I think her name was Atkins or Perkins, but if I were to decide, I must say neither sounds incorrect.
I think her name was Atkins or Perkins, but if I were to decide, I must say both sound likely.
"not either" might seem like shorthand for "neither" but "neither" is usually applied without alternatives: "neither X nor Y", period. "not either" will be usually in construct of alternative: "not (either X or Y) but Z" - and even then this construct is fairly rare.
Syntactically, OP's first alternative (Could not you help him?) is more "correct", in that it places the negating not closer to could - the element it most applies to. But idiomatically today we almost always contract that to...
Couldn't you help him?
...even in "formal" contexts. But people often think of such contractions as "informal", and precisely because everyone normally makes the above contraction, the full form as given by OP tends to sound "strange" to the native ear. Thus, when looking for more formal phrasing we're likely to think in terms of discarding the contraction, but we avoid OP's #1 and go for #2 because it doesn't sound quite so odd.
In fact, given a formal setting (lawyer questioning witness, say) there's nothing at all unusual about...
"Is not this the murder weapon, Professor Plum?"
"Could not the gun cabinet have been left unlocked, Colonel Mustard?"
"Was not the dagger kept with the other cutlery in the kitchen, Mrs Peacock?"
OP's idea that the position of not determines whether it's a genuine enquiry or a complaint isn't quite right. What matters is that a complaint would stress the word not - which you can't do if it's contracted. You want that stress because effectively all the other words in the utterance apart from could and not simply describe the current situation (although less common, in most contexts it's possible to stress could instead of not when the intention is to complain about something rather than ask if it's true).
Note that it's not always possible to distinguish a question from a request. For example, given just...
"Could you not do that?"
...it can often be effectively impossible without further context to establish whether the speaker is asking you to stop doing something (complaint) or asking whether you're unable to do it (question).
In short, context and intonation are crucial in such negated constructions. Consider, for example,...
"Could you not do that?"
...which might mean...
1: You're doing it right now, and I want you to stop
2: It's what you plan to do, and I'm asking if you will (or are able to) change your plan
3: It's not what you plan to do, but I'm asking if you will change your plan
(and probably other possibilities)
If it's not obvious how #2 and #3 apply, consider "Could you not work tomorrow?", which could either be asking you to take an unscheduled day off, to work on a day when you normally wouldn't.
Best Answer
This is a matter of style. Both constructions are correct. But it's better to not split the infinitive(as what I've been taught in school). Ngram also shows that not to do is more common than to not do.