In Swedish, the verb has to go in second place in the sentence (in main clauses that are not questions). But we like to structure information by putting the 'topic' (the thing we're speaking about or the starting point of what we're saying) first in the sentence. So once we put the object first for that reason, the subject will have to go third because the verb stays in second place no matter what.
Yes, that's why we translated it the way we did (the main translation on top of this page). But the idea, in terms of information structure, it's more like you put it, or 'That is something neither you nor I know'. Starting out with that or it isn't as natural in English in this case, but in Swedish it's pretty common.
It means roughly 'I don't know that and neither do you'. This is a very natural sentence in Swedish, we like to stress different parts of the sentence by using constructions with det, but since English doesn't work the same way, the English translation needs to have a different word order.
The neither-nor; either-or is taught in UK schools, or at least was when I was at school. If we hear someone use this incorrectly, we would assume that they're either uneducated or lazy. Having said that, it is a very common mistake among native English speakers because, I think, using correct grammar comes second to being able to communicate your thoughts.
A swedish word that may get its meaning changed is björntjänst. (Lit. Bear favour)
Att göra någon en björntjänst (to do someone a bear favour) means to help someone in a way or with something that is bad for that person in the long run.
Some (young) people use it with the meaning a huge favour nowadays. In doing so, I think that they are doing the Swedish language a björntjänst, because we might 'loose' a very good meaning of a useful word.
Btw. Lately I've said 1+1=3 as an expression to say that the sum is bigger than the parts separately.
Or as the Swedish poet and Nobel prize winner Tranströmer put it the other way around:
"ett kilo vägde 700 gram inte mer". Ur Dagsmeja, 1962
Neither you nor I know him, would be: (Preferable) Varken du eller jag känner honom. (but in some cases this could sound ok:) Honom känner varken du eller jag. He doesn't know you or me. would be: Han känner varken dig eller mig. Note the difference of meaning in the last sentence. Han means he, honom means him. Hon means she, henne means her.
In English, "neither" is singular, and somewhat abbreviated; the complete form would be "neither one". So "Neither you nor I knows it" is the precisely correct answer. I'm not advocating that "...know it..." should be removed but I believe that "...knoes it..." should be included in the list of correct answers.
It doesn't matter whether "you" or "I" or "we" or "they" is in the sentence; the subject is "neither" and it is singular, so the verb has to be "knows."
Take out the non-essential words and you have "Neither knows" or "Neither one knows." The other words throw people off and make them think the subject is plural, but it is actually singular, and the verb must match the subject, of course.
My English teachers would all disagree with you, I think. "Neither" or the implied "Neither one" is the subject, and it is singular, therefore, the verb has to be singular. This would apply even when plurals are involved, such as: "Neither the students nor the teacher understands." "Neither of us is going." "None of the parents approves." "None of us is willing to go." "Not one of us is willing to go." "Neither parent is happy about it."
People make the mistake in everyday speech sometimes, but the correct sentence should definitely be included as a correct answer for this exercise.
All of your examples use a singular, though, and not the same construction.
Quoting Oxford Dictionaries:
In the constructions either. . . or and neither. . . nor, you can use either singular or plural subjects, which should be matched to singular or plural verbs, respectively:
* It’s one of the best songs that either he or anyone else has composed.
* Neither the politicians nor their advisers are able to determine these outcomes.
Edit: Although that said, I kind of forgot which sentence we were talking about. :) In the case of "neither ... nor", the subject most closely connected to the verb affects the agreement. Hence, the formally correct sentence is "Neither you nor I know that". For a different sentence, it would have been "Neither you nor I am."
"neither" is a conjunction when coupled with "nor", unless used as an adverb.
- Conjunction: Neither we nor they like to sing.
- Determiner: Neither of them likes to sing.
- Adverb: They don't like to sing, and neither do we.
It can be a pronoun as well but that isn't really relevant here.
In addition, from what I can find, there is great variation in how natives use these. I would consider taking any "rule" on the subject with a grain of salt.