Learn English – Is “I have had to take a leave” correct

The leave them kids alone part is definitely ungrammatical, as is the first chorus line We Don't Need No Education (the former should be those kids, and the latter uses the much-reviled double negative).

Per this answer on a related ELU question, using "them" as an "article" is non-standard (but not uncommon in speech, particularly with the less well-educated, or in "facetious" usages).

There's nothing ungrammatical about leave us kids alone - if there were, you wouldn't expect to see written references to things that might help us writers, for example.

In such constructions, the [optional] "class/category name" after "us" applies to all of us - it doesn't select kids or writers from a larger group. The context inherently implies that "we" are all kids, writers, whatever.

Both "I won't be needing that" and "I don't need that" are grammatical.

The problem with your teacher's logic is that "need" is not a modal in either of those. Two easy-to-detect signs of modality are:

• They have the syntactic properties associated with auxiliary verbs in English, principally that they can undergo subject–auxiliary inversion (in questions, for example) and can be negated by the appending of not after the verb.
• They do not inflect (in the modern language) except insofar as some of them come in present–past (present–preterite) pairs. They do not add the ending -(e)s in the third-person singular (the present-tense modals therefore follow the preterite-present paradigm).
  _Wikipedia

For the first point I listed, it should be pretty obvious that not does not follow need in "I don't need". For the second point, since people say "he needs...", we know it's not modal in this sense either.

"Need" specifically is a semi-modal. Your sentence needs modification were it to use need as a modal:

I need not have help.

According to the statistics here, the modal form of need is not used very often in American English (which is why I think it sounds old fashioned). A COCA search for BE needing (capitalizing "be" means it matches all forms of the verb, such as be, were, 're, etc.) returns 356 results, so I think it's safe to say that educated speakers do use "will not be needing" and similar (I would).