do italians never say 'lei è nostra vicina'?
The only exception is not using the article is when you're talking about your family members. "Mio padre", for instance.
Cento percento, here, in duolingvo i meet some examples where definite article was omitted in case not of family members, but just humans.
The article is required in this case.
Is there a way to say she is a neighbour of ours in Italian, using una ?
I wondered this, too. Certainly in English "the" implies that we have only one neighbor.
How frequently does an Italian word have both a masculine and feminine version? Is this only and always when referring to people?
for people and animals too...il gatto...la gatta
Some animals, like gatto / gatta. But some animals are neuter-generic, like l'orso.
How about Lei è la vicina nostra?
Word Reference has only "'vicino" as a noun, no feminine noun. However as the adjective "near" or "close" there are listed all four endings.