New Fowler's Modern English Usage devotes a full half-page to neither/nor singular/plural, pointing out that there is now less tolerance for the use of plural verbs after singular subjects than there was in the 18th and 19th centuries.
Purists will expect a singular verb where both elements are singular, a plural where both are plural. Where there is one of each, he suggests avoiding the construction completely in writing, and to follow instinct in speech.
In the case of the sisters, neither of the sisters is directly equivalent to neither sister, and the verb should be singular.
In the case of the request to chat with you and your dog, the problem is not singular/plural but which person of the verb to use - and of course you picked a verb with different first/third person forms
1. Neither my dog nor I am available for chatting. - sounds dreadful
2. Neither my dog nor I is available for chatting. - sounds dreadful
3. Neither my dog nor I are available for chatting. - sounds slightly better, but not good.
4. Neither my dog nor am I available for chatting? - sounds dreadful
So, what's the answer?
The answer is to avoid this construction completely in writing and if you find yourself already down that sentence in speech, use (3).
1) Neither of the sisters knows the truth, and
2) Neither of the sisters know the truth
I would say and prefer the first, but not really complain about the second one. If these appeared on an exam or in homework, I'd say you have to choose the first, but I'd lament the waste of effort involved in testing such a point. Of course you could just say: 'Neither sister knows the truth'.
When it comes to the 'neither... nor...' sentences, I like elroy's guideline in the other threads, that the verb agrees with the closest subject, since that's how our brains work any way. (That's why 'none' is becoming plural: in 'none our friends is/are coming', 'none' is singular but we're influenced by the plural 'friends', so more and more people say 'are', even though 'is' is more logical.)
As for the others:
1. Neither my dog nor I am available for chatting - sounds OK to me (well, except for the chatting bit)
2. Neither my dog nor I is available for chatting - sounds dreadful.
3. Neither my dog nor I are available for chatting - is OK as a compromise.
4. Neither my dog nor am I available for chatting - sounds horribly confused.
The last one breaks the pattern of neither/nor sentences. When you say: "neither A nor B...", A and B should be grammatically equivalent, either noun phrases or verb phrases or adverbial phrases or prepositional phrases or whatever. You can't invert the verb with one of the subjects in this way. It makes it sound like 'am' is one of the options being negated, an equivalent to "my dog", and then there's no verb left over for the rest of the sentence.
("Nor am I" is fine as a tag, of course:
A: I'm not going to school today.
B: Well in that case, nor am I.)