Should Not Need Not?

"Should not" indicates that it is bad to do a particular thing. For example, you should not drive a car too fast, because driving too fast is dangerous. "Need not" indicates that it is not bad not to do a particular thing (in other words, the thing is unnecessary).

Should not or do not have to?

Mustn't means something is prohibited or it is not allowed. Don't have to means there is NO obligation to do something. You are not required to do something, especially if you don't want to.

Where do we use need not?

Need is used to express necessity or obligation to do something, generally in an interrogative tone. The negative form needn't (or need not) is also used to express necessity or obligation; but in the negative sense—i.e., absence of necessity or obligation to do something.