You have two finite verbs, lives , and is . So you have a compound predicate, requiring a conjunction: The woman lives next door and is a doctor. "Finite" has to do with time.
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TerI'm having trouble explaining "why" a sentence can't have the following structure.
Wrong: The woman lives next door is a doctor.
When the subject of a relative clause is the pronomialized element, it cannot be omitted.
The woman [she lives next door] is a doctor. > The woman who lives next door is a doctor.