Learn Italian: Sono innocente, non stavo rubando il gioiello! - I am innocent, I was not stealing the jewel!

To say you "were doing" something in the past, Italian uses the imperfect tense of stare (here, stavo for "I was") plus the gerund of the main verb. For rubare (to steal), you drop the -are and add -ando to get rubando.

Where exactly does the non go when using this two-part verb?

The non always goes immediately before the conjugated helper verb (stavo). You cannot split the two verb parts, so saying something like stavo non rubando is incorrect.

If a woman is speaking, would she say Sono innocenta?

No, she would still say Sono innocente. Adjectives that end in -e in their singular form do not change based on masculine or feminine gender. They only change to -i for plurals (siamo innocenti).

Why is it il gioiello and not lo gioiello?

Even though gioiello has a lot of vowels, it starts with a soft "g" sound (like the "j" in "jewel"). Since it starts with a regular consonant sound, it takes the standard masculine article il. The article lo is saved for masculine words starting with z, y, gn, ps, or s plus another consonant.