Use of "as ..., so ..." in negative forms

Sentence (1) sounds awkward because the two sentences have the same lexical verb live. Typically the lexical verb is left out of the so... part if it is the same as the the one in the as... part. It could be changed to:

As fish can't live without water, so can't people without air.

The subject-auxiliary inversion is also optional in the so... part, and the sentence seems to read better without it, even if the lexical verb is repeated.

As fish can't live without water, so people can't (live) without air.

As for (2), the so from (1) is optional in these constructions (CaGEL p1539), so removing it is a valid choice. Neither is a connective adjunct in clause structure that triggers subject-auxiliary inversion(CaGELp 1308) appearing in constructions like:

If you don't complain, then neither will I.

This would keep the as... part as an adjunct of manner and have pretty much the same effect as (1).

Sentence (3) has no grammatical issues. Nor is used non-correlatively as a variant of or in negative contexts, and allows subject-auxiliary inversion in which case it is no longer replaceable by or (CaGEL p1309). However, the meaning here is not quite the same as (1) because it loses the adjunct of manner, the sense of in the same way.

Sentence (4) is clearly grammatical, and very close in meaning to (1). More here is in modifier function, but the usual degree meaning has effectively been lost. We don’t interpret (4) as a comparison between the degree/extent to which fish can't live without water and that to which people can't live without air (CaGEL p1133). It can be restated without a degree modifier as:

Fish can't live without water, just as people can't live without air.

What is wrong with sentence 1 is that the not is in the wrong place.

A grammatically sound rephrasing is

As fish are not able to live without water, so are people not able to live without air.

Now that we've identified what's wrong with the sentence, we can try doing the same thing with can:

*As fish cannot live without water, so can people not live without air.

Unfortunately, this doesn't look correct to me. I think the problem is that we are so used to seeing cannot written as one word that it doesn't look right when we split it into two, as we are forced to do by this sentence. A similar sentence

As Ben wouldn't eat his breakfast, so would Beth not eat her dinner,

sounds fine to me, while

*As Ben wouldn't eat his breakfast, so wouldn't Beth eat her dinner,

sounds wrong.