Let's simplify it down by turning one of the "multiple choices" into a single value. Like we could say, "Neither the van nor the truck is red." That's pretty clear and unambiguous. The van is not red and the truck is not red.
"Neither" is basically short for "not either". So let's go back to "Neither the van nor the truck is neither red nor yellow." Let's drop the yellow. Now, "neither" is a "negating word", so if there was only one option, it would have to be replaced with "not" or something similar. That would leave us with the analogous sentence, "Neither the van nor the truck is not red." I think that means that both ARE red, but it's a contorted sentence, at best unclear.
So going back to the two-by-two case, I think it follows that if you negate both sides, you turn it into either one of those "double negative equals a positive" cases or just a jumbled mess. You have to make just one side or the other negative. That is, either:
Neither the van nor the truck is either red or yellow.
Or
Both the van and the truck are neither red nor yellow.
Or maybe better still:
The van is neither red nor yellow. The same thing goes for the truck.
:-)
That isn't double-negation, that's single negation. Double-negation would be this:
That book does not have no pages.
Sentences like the above are correct in many languages and are used in many non-standard forms of English, but are not correct in standard English.
You may be confused by the word any, which is a negative polarity item. The word any is not itself a negative, but it is required in place of no or some when the main verb is negative.
Best Answer
Your two exact correct sentences are explained well on the Cambridge Dictionary website, which I found by using Google. It's a very good read, and I learnt some things from reading it despite English being my first language.
Taken from the website, "when a clause with neither or nor is used after a negative clause, we invert the subject and the verb after neither and nor." Looking at your two first sentences:
"He" and "had" were inversed after a negative clause in your first sentence, and "we" and "did" were inverted in your second sentence.
Whilst there are other situations in which "neither" and "nor" can be used, none of them apply for your three incorrect sentences.
"Nor" either follows a negative clause as explained above, or it follows "neither." Neither are the case for your second and third sentences. When the criteria for "nor" are not met, "or" should be used instead:
"Neither" can be seen as a combination of the words "not" and "either." In your incorrect first sentence, you have already used "not" as part of "haven't;" this invalidates using "neither." To correct the sentence, you would either write:
(Removed "not" from "haven't", so it now makes sense to use "neither." As "neither" has been used, it is also appropriate to use "nor.")
– or you would write:
(Removed "neither" and "nor.")