Neither of your proposed options sound correct. Neither and nor are often used together, for example:
He ate neither meat nor bread for supper.
The meaning of which is
He did not eat meat for supper, and he also did not eat bread for supper.
Now, in the above examples, the neither/nor construction is idiomatic because you're referring to two things which he did not eat for supper. That is
He ate [x] for supper.
He ate [vegetables] for supper.
He ate [neither meat nor bread] for supper.
As you can see, both of the items referred to by neither and nor are things which were not eaten for supper. So it makes logical sense to pair them this way. (I'm not saying this is the only way in which neither/nor can be used, but it's common, and hopefully will help you understand the problem with your sentence.)
So now we return to your examples:
I am not your slave, nor do I have to follow your advice.
I am not your slave, neither do I have to follow your advice.
Neither of these sound right, and here's why: not being a slave and not having to follow someone's advice are two completely separate ideas. If they were both things I am not (your slave), things I don't have to do (follow your advice), or, as mentioned previously, things I didn't have for supper, then you could combine them with neither/nor. For example:
I am neither your slave nor your pet.
The category is things I am not; I am not your slave and I am not your pet. This is fine.
Now, it is also possible to change I do not have to follow your advice (things I do not have to do) into I am not required to follow your advice (which now makes it a thing I am not). In that case you could use neither/nor:
I am neither your slave nor required to follow your advice.
Now the construction makes sense because both items fall into the same category. This is actually a useable sentence for your purposes; you could say it and you would be understood just fine. It does come across as a bit formal, though, and I don't think you're likely to hear it in casual conversation.
So to select the wording which is most likely to be useful in conversation, forego neither/nor entirely since your two items are not of the same type. Instead say this:
I am not your slave and I don't have to follow your advice.
The and combines two separate but true statements. Simple and to the point!
The phrase no one of them is "grammatical" (taking that to mean that it is both idiomatic and meaningful), but it is not always the same thing as none of them. It is an emphatic form employed to distinguish a proposition concerning a number of entities taken singly from propositions concerning the same entities taken in combination.
Consider the difference between these two propositions concerning three people, Adolphus, Benjamin and Carloman, suspected of murdering Dionysius.
None of them killed him. This is taken to assert that all three are innocent of the murder.
No one of them killed him. This implies the possibility that two of them—A and B, or A and C, or B and C—or all three, conspired to commit the murder.
So it might be perfectly proper to say of three utterances that "No one of them is grammatical" if you intend to propose that some combination of the utterances is grammatical.
Best Answer
In essence, all three sentences mean the same thing. However, if you want to delve into the nuances, (2) and (3) give greater emphasis on the objects (the letter and postcard).
However, unless one was asked to carefully examine these three sentences, they would see no difference between them.