Learn Spanish (Spain): Mi madre, la médica, es la propietaria del edificio. - My mother, the doctor, is the landlady of the building.

You are right that we drop the article when stating a profession directly after the verb ser (to be). However, here la médica isn't following ser. It is an apposition (a descriptive phrase placed right next to a noun to give extra information). Because you are literally saying "My mother, the doctor" as an aside, the definite article la is required.

Is médica the standard word? I was originally taught to say la médico for a female doctor.

While la médico was common in the past because many professional titles traditionally only had a masculine form, the language has evolved. Today, la médica is the standard, fully accepted way to refer to a female doctor in Spain.

Could I use dueña instead of propietaria in this sentence?

Yes, absolutely. Both dueña and propietaria translate to "owner". Propietaria sounds slightly more formal or legal, similar to "proprietor" in English, but they are practically interchangeable in everyday conversation.

Why is propietaria translated as "landlady" here instead of just "owner"?

The literal translation of propietaria is indeed "owner". It translates to "landlady" in this English sentence purely because of the context: she is the owner of a building (edificio), which usually implies she rents it out. If she owned a car, she would simply be translated as the "owner".