I think the sentence structure in my sentence number 1 is correct.However as Wendikidd has pointed out the sentence does not make any sense.But if it were making sense,then 'nor' can be used without neither.This is what I have understood.

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!

Nor can be used separately from neither, for example:

I won't go on social media or message boards. Nor will I read comments posted beneath online newspaper articles.

It's true that your sentences sound a bit weird; I think the first one can be good. And the second one maybe too. But there are better ways to write them.

1. I am not your slave. Nor am I forced to follow your advice.

2. I am not your slave, I don't have to follow your advice either.

However I have read on the Internet the following construction pretty often so people use it although it may be not totally correct grammatically:

I am not your slave, neither am I forced to follow your advice.

I have to say that I am not a English native speaker although I have lived in the USA for quite a long time and I keep studying English.... so I can be wrong.

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.

Both sentences are perfectly correct, although in the second one a semicolon might be preferable.

The Cambridge Grammar of the English Language, under the heading "Nor with subject-auxiliary inversion" (p1309), gives examples such as:

• He didn't attend the meeting, nor was he informed of its decisions.
• He was one of those people who can't relax. Nor he did he have many friends.

(Note: The second example in particular disproves the notion that the two clause linked by "nor" need to be closely connected.)

CGEL goes on to note that "neither" "could replace" "nor" in these sentences. However, "neither" is properly an adverb, so in these sentences it would work better either at the start of a sentence or after a semicolon, whereas "nor" can begin a sentence or can simply follow a comma.

The OED has interesting examples too (these ones are all 20th-century citations):

• They were the worst, most difficult, most agonizing days and nights of my life. Nor did I leave with a clean slate.
• The Arabs were poets, dreamers, fighters, traders; they were not politicians. Nor had they found in religion a stabilizing or unifying power.
• If you cannot indict a nation, neither can you fully describe a nation, environmentally or spiritually.

I am not your slave and I do not have to follow your advice.

This, too, is perfectly correct.

The words "either" "or" and "neither" "nor" are used together in a sentence.

For e.g. : Either Mary or Joe will visit their grandmother today.

So, here any of them will visit their grandmother today.To make a choice between two options , the pair of "either or" is used

For e.g. : Water was neither cold nor hot.

So, here, by using the pair of neither nor, we can say that none of them is true. It is understood that water is warm. It is neither cold nor hot.