"How the Hell would I know?" 
I know which of my coworkers are g
ay the same way I know which of my coworkers are straight. I've met their spouses at parties and I've heard them use gender-specific pronouns when referring to their SOs in everyday conversation.
CW’s questions ARE out of line. OP continued to engage in discussion about the topic, which was a mistake. People like this CW need to be cut off quickly and firmly.
Other than hello, goodbye, thanks, excuse me, and mandatory interaction I would not interact with this CW.
In order to address this and other similar posts I need to tell you two things about my job. First, the organization I work for deals with social, political, and religious issues. There are frequent candid conversations about things like homosexuality, poverty, disease, and discrimination and their place in contemporary American society. (While I am not directly involved in these conversations, part of my job is helping participants find relevent background information. CW is both a participant and a helper like me.) So while these sorts of discussions don't
have to get personal, the nature of the subject matter can make it hard to stay completely clinical.
The second thing is that a sense of community is very important to the success of the organization. Since I'm not directly involved in the aforementioned conversations I can remain aloof to a point, but not nearly as much as if I were, say, the groundskeeper.
So I guess the takeaways from this are 1) the kinds of political discussions that would be verboten at other places of business are not uncommon here, and 2) me having a strict 'business talk only' policy is not feasible. Obviously CW was still way out of line to ask what she did, but maybe this will help you understand the context a little better.
Oh, and as far as flat-out avoiding her as some have suggested-- That's not possible for both work reasons (part of my job is assigning her tasks and part of her job is receptionist-type things like bringing me my mail) and geographic reasons, as she has to walk past me to get to her car and I have to walk past her to go anywhere in the building outside our department. Silver lining: Our desks are nowhere near each other (not even within earshot

), and she shares her job with about half a dozen other people, only one of whom is ever on duty at any given time.