Learn Italian: Sei permaloso? No, non lo sono. - Are you touchy? No, I am not.

This is the pro-predicative lo. In English, we can just say "No, I am not," leaving the sentence hanging. In Italian, the verb essere (to be) usually needs a predicate to complete it. The pronoun lo steps in to replace the adjective permaloso (touchy) so you don't have to repeat it.

If a woman is answering, does she say non la sono?

No, she would still say non lo sono. When lo is used to replace an adjective (or an entire concept), it acts as a neutral pronoun and is completely invariable. It never changes to match the gender or number of the person speaking. She would, however, be asked "Sei permalosa?".

Could I just translate the English literally and say No, non sono?

No, that sounds grammatically incomplete to an Italian ear. While English allows you to drop the adjective after the verb "to be," Italian requires you to either repeat the adjective (No, non sono permaloso) or use the pronoun lo to represent it (No, non lo sono).

What is the exact nuance of the word permaloso?

It translates to "touchy," "oversensitive," or "quick to take offense." It describes someone whose feelings are easily hurt or who gets defensive over small jokes or mild criticisms.