Learn Italian: Nostra suocera è un medico. - Our mother-in-law is a doctor.

Normally, yes! In Italian, you usually need a definite article before a possessive. But there is a special rule: when you talk about exactly one family member, like suocera (mother-in-law), you drop the article. That's why it is just nostra suocera.

If the mother-in-law is a woman, why does the sentence use the masculine un medico?

The word medico is historically masculine and is commonly used for both male and female doctors in everyday Italian. If you wanted to use a specifically feminine word for a doctor, you could say una dottoressa instead.

I learned that we shouldn't use words like un or una with professions in Italian. Is this an exception?

You have a good memory! Italians often drop the 'a' or 'an' when stating a job, so è medico is perfectly correct. However, adding un is also valid and very common in everyday speech. Both ways are fine!

If I were talking about more than one mother-in-law, would I still drop the article?

No, you wouldn't! The rule about dropping the article only applies to singular family members. If you make the noun plural, the article comes back: le nostre suocere.