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!
Best Answer
I don't know how you arrived at this sentence, but it is wrong.
You could just say "I should come tomorrow." but as you are looking for a time you wouldn't get much of a response because it is not a question.
"Should I come tomorrow?" gets you a step closer and gives you an answer either "yes" or "no".
Now, "At what" is a verb phrase, and it needs a direct object. In your first sentence, you are missing the direct object. In your second construction the word "time" is your direct object.
Essentially you want to ask yourself "At what, what?" meaning, what am I looking for? In this case (since it's a question) What so I want someone to respond with? Do I want them to respond with a place, a time, or something else? Because you want them to respond with a time you need to indicate that you are looking for a time as an answer.
In your first construction there is no indication that you want a time. "At what what?" - there is nothing. "I should come tomorrow" (I believe) is a prepositional phrase, which modifies "At what". Since there is no other words in the sentence "At what" is left hanging and you don't know what is being asked for. So you need the word "time" there so that you have a complete sentence.
So your correct answer is:
However most people would say: