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.
Only your example 1 is correct.
The verb "to be located" is the passive form of the less common verb "to locate" (in the corresponding active sense of deciding upon placement), so it is indeed a verb phrase, and your instinct is correct that it requires inversion when phrased as a question. However, when inverting word order to form a question, as in example 1, only invert subject (apartment) and helping verb (is), never the main verb (located).
However, the answer to the question, which is not inverted, would be constructed as
Your apartment is located here.
By way of comparison, when asking the question,
Where is she going?
You would never ask,
Where is going she?
But you would answer,
She is going there.
Best Answer
It's a case of split or cleft infinitive. Some people object to separating 'to' from an infintive usually by putting an adverb such as either between the to and the infinitive. But there's no hard-and-fast rule about it; you can go either way.
But it's very common not to split an infinitive. So the OP's first sentence is better than the second one.