Learn Italian: Non sono per niente stanco.

Both are used to emphasize a negative statement, but they have different flavors. Non sono per niente stanco emphasizes the degree, meaning "I am not tired at all." On the other hand, Non sono mica stanco is used to correct a false assumption, meaning "It is not like I am tired" or "I am not tired, in case you thought I was."

Do you always need to use non with per niente?

Yes, when used in a full sentence with a verb, Italian requires this double negative structure. The non goes before the verb, and per niente goes after it. However, if you are just answering a question without using a verb, you can just say Per niente! (Not at all!).

Could I use per nulla instead of per niente here?

Absolutely. Non sono per nulla stanco means the exact same thing. Both per niente and per nulla are very common and interchangeable ways to say "not at all" in Italian.

Would this sentence change if a woman were speaking?

Yes. Because stanco (tired) is an adjective describing the speaker, it must agree with their gender. A male speaker says Non sono per niente stanco, while a female speaker would change the ending to say Non sono per niente stanca.