didn't that used to be spinster? and isn't now career woman? do we have any better choices?

Well, I know the word spinster fell out of use some time in the '70s. Career woman is *never used here. You can say a woman has a careet in sales or some other job but she is never refered to as a career woman.

A woman that is not married is 'single' (célibataire in French)

We don't have a word for the residence of a woman living alone like men do with bachelor appartment.

The term 'bachelor' in French refers to an appartment (flat) that has a bathroom, a bedroom, a kitchen and a living room. We call that a "three and a half".