need not come vs don't need to come

Question

1. You need not come unless you want to.
2. You don't need to come unless you want to.

The second sentence is right but first one is wrong. Why?

My second question is the necessity of "to" after "want". Would this be wrong:

3. You don't need to come unless you want.

Answer 1

The second sentence is right but first one is wrong.

According to whom? Both sentences are perfectly grammatical. In fact, according to Google Ngrams need not come is substantially more common than do not need to come, though the former is losing ground and the latter gaining.

In general, English verbs can be either regular or modal. A regular verb uses the auxiliary do for negatives and questions:

• You sing, when turned into a question, becomes Do you sing?
• I teach, when made negative, becomes I do not eat seafood.

Modal verbs, on the other hand, are words like can, might, should, etc. make no use of this auxiliary do:

• He might, when turned into a question, becomes Might he?
• We could, when made negative, becomes We could not.

Need is one of a tiny class of English verbs called semi-modal. This means it can operate either as a regular verb or a modal. It is by far the most common of the semi-modals; dare is a distant second. As semi-modals, verbs like need and dare can use do or dispense with it, more or less at will. Hence all these are grammatical:

• Need I pay all my term fees upfront?
• Do I need to pay all my term fees upfront?
• I need not listen to such drivel.
• I do not need to listen to such drivel.
• I don't dare talk to that handsome guy in our French class.
• I dare not talk to that handsome guy in our French class.
• Dare I eat a peach?
• Do I dare to eat a peach?

So your first two sentences are both correct. As for the to at the end of the second, it would probably be okay to leave it off in speech, but would be considered odd in most writing. In certain contexts, leaving off that to can alter the meaning rather drastically:

Don't kiss anybody you don't want.
Don't kiss anybody you don't want to.

The second sentence is rather more innocent than the first.

Answer 2

Modal verbs can be complex and seemingly irrational. However, I think I can explain this one.

"Need/need to" is a semi-modal, not a full modal. In the first sentence, it acts as a full modal (with the negative particle "not" added directly to the modal verb), where as in the second example, you need the auxiliary "do" to create the negative.

This changes the position of "not" in the sentence, and as word order is particularly important in English, it changes the meaning. Compare:

You needn't come = what you need to do is not come.

You don't need to come = there is no need for you to come (but a possibility to come still exists)

You can apply the same to must/mustn't (full modal) and have to/don't have to (quasi-modal) to understand why the affirmatives appear to mean the same as each other, but the negatives mean something different from each other.

Anyway, if what you need to do is NOT come, then the adverbial phrase "unless you want to" is clunky and awkward and perhaps, as you said, wrong.

As for the third sentence, well...

You need the word "to" for the omission of "come". "to" is the identifier that the word "come" is missing (to avoid repetition). In the sentence You can come if you want. there is no confusion, and so it is perfectly acceptable, as is You can come if you want to.