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.
'Two sides of the same coin' does not quite mean what you describe.
two sides of the same coin - different but closely related features of one idea
It essentially means that two things are the same. I might use it in a context where someone is describing someone else as both 'lazy' and 'messy' and in response I could say that those are 'two sides of the same coin' and thereby suggesting that they are one and the same, in this case suggesting that the person is messy because they are lazy.
As far as I can tell you want a pithy phrase to describe two things that are good together but not necessarily either good or as good apart.
I'd suggest a simile in this situation. A very common type of phrase is to say:
[Something] without [something] is like [something else] without [something else].
Currently on the London Underground for example there is an advert that says something along the lines of 'a woman's hair without product x is like rock without roll' (the latter part usually a ridiculous separation for humour value). There's no standard phrase used here but it's an opportunity to be creative.
A couple more examples:
A man without ambition is like a bird without wings (from a 1908 business magazine)
A house without books is like a room without windows (a proverb that goes back even further)
Best Answer
To me, "of them" comes across as slightly wrong, because it's partitive. Consider:
A quantity of something implies part of a whole: Susie ate a part of the whole set of six apples. But in the second sentence, which isn't partitive, one would not ordinarily say "she ate two of them".
Another example:
The of construction in the second sentence clearly suggests that I have more sisters than two, whereas its absence in the first clearly indicates that I have only two sisters.
Similarly with your favorite colors, or the cars of your dreams: you have a fixed quantity, and you're speaking of the entire set. So you might say "Two of my dream cars are a BMW and a Mercedes" if you mean that you also dream about a Porsche and a Jaguar. However, if the first two are the only ones you dream about, you'd simply say "I have two dream cars, a BMW and a Mercedes." Therefore, when answering the hypothetical question, the elliptical form to use would be simply "I have two," not "I have two of them."
As for "there are two": it's better than either of the "of them" alternatives, and there isn't all that much to choose between it and "I have two."