This is one of those areas where a prescriptive grammar rule is not terribly useful, because native speakers so regularly flout whatever rule you care to come up with. Instead, let me list which ones sound OK to my ear, and which ones do not. (I'm slightly reordering your examples to make my point better.)
1 Neither sound natural.
7 Neither sounds natural.
Because there is nothing between neither and the verb in this case, there's really no choice but to make the verb agree with the number of neither, which is singular. So #1 sounds wrong, while #7 sounds fine.
2 None sound natural.
8 None sounds natural.
Same situation here, except (at least to me) none on its own is plural (like zero would be), so #2 is OK but #8 sounds wrong.
3 No one sound natural.
9 No one sounds natural.
This one's pretty clear cut: one is singular, so the verb must be sounds, i.e. #3 is wrong, and #9 is correct. (I do want to add that this is a pretty unnatural sentence unless you're using no one as a synonym of nobody. If you're talking about example sentences in a book, don't use "no one".)
4 Neither of them sound natural.
10 Neither of them sounds natural.
Because of the "of them" that comes between "neither" and the verb, this can go either way: you can either use sounds, making the verb agree with neither, or you can go with sound, making the verb agree with them. So both #4 and #10 are OK.
5 None of them sound natural.
11 None of them sounds natural.
Example #5 is clearly fine -- the verb agrees with both none and them -- but, surprisingly perhaps, so is #11. This is one of those strange cases where, because of the intervening words between none and the verb, my ear no longer cares that none is supposed to be plural. Bottom line, #5 is the better choice, but #11 isn't wrong, either.
6 No one of them sound natural.
12 No one of them sounds natural.
I don't know if I'd recommend either of these. I think my problem is semantic rather than grammatical — as I noted above, no one really just functions as a synonym of nobody, and you don't say
"nobody of them". In a context where I wanted to stress that no single one sounded natural (perhaps there are combinations of items that sound natural, but none of them work on their own), I can perhaps see #12 sounding OK.