most of the dictionaries state that neither cannot connect sentences, as in Oxford Dictionary:
neither | Definition of neither in English by Oxford Dictionaries
‘he didn't remember, and neither did I’

we can see that the writer used another conjunction, namely and to chain the 2 sentences. Therefore the writer thinks the word neither cannot have done the job.

However, in Cambridge Dictionary English Grammar Today it states that
He hadn’t done any homework, neither had he brought any of his books to class.
source: Neither, neither … nor and not … either - 當代英語語法 - 劍橋線上詞典
In this case, it is apparent that the writer believe the word neither could be used to connect 2 sentences.

So, in standard English, could the word neither serve this purpose or not?

 

I like the first example better.

In the second one there should be a semi-colon at least to mark them as two separate clauses within a single sentence.

 

I agree with boozer. In AE, in the second example, I would change one of these things:

[1] I would use nor instead of neither, or
[2] I would use a semi-colon instead of a comma.

 

I think the following sentences are better.

He hadn’t done any homework. Neither had he brought any of his books to class.
He hadn’t done any homework; neither had he brought any of his books to class.
He hadn’t done any homework, and neither had he brought any of his books to class.
He hadn’t done any homework, nor had he brought any of his books to class. (Nor is a coordinating conjunction)

 

The "neither had he..." construction sounds very old-fashioned to my ear. It's common in the King James Bible. "They toil not, neither do they spin..." Matthew 6:28.

 

Using "neither" as a conjunction sounds very old-fashioned to me too. Even "nor" is rarely used in modern AE.

 

Or BE. From over 200 years ago "Water, water, everywhere, nor any drop to drink." (Ancient Mariner, Coleridge)