I work with a gay man and he uses the women's restroom instead of the men's. Does he maybe identify himself as a man or as a woman?

Well, no one can answer that question except your coworker. Most of the time, it seems to me that when trans people go to the restroom that "match" their gender, they don't do it before presenting themselves in their gender. Because we're often very scared of what people could say and/or do to us, it can be very dangerous for our safety... But maybe that person do. And maybe they're not trans at all but just feel more safe in the women's restroom. There are lots of reasons for that. For instance, they could have had bad experiences in the men's, or they could use the stall and none was available in the men's... or because women's restrooms are cleaner lol, or maybe they didn't pay attention to the sign on the door... If you think it's very important for you to know (I don't really see why it would be but, well..), and can't resist asking the question, maybe keep it neutral, without exposing your ideas. For instanceyou could just say that you noticed they used the women's restrooms, and you were wondering why, if they're ok to tell you... And don't insist if you see that seems to bother them to reply. But really, and that's the most important here : if this person wants to tell you something, they will. If they don't, it's probably none of your business anyway^^

Using the women's restroom sounds like a signal that he identifies as a trans woman. However, it's better not to bring it up and let him/her handle it: whatever it means, if he/she wishes to tell someone, it's his/her choice to do it. You can just leave it for now, since everyone's identity is a personal matter until they decide to tell it to others.

Yes, it is very much possible. From my experience, I had a lesbian girl friend who used the women's restroom, but people always saw her as a man (physically) and everyone would feel uncomfortable when she used the woman's bathroom. One day, she started identifying as a man and using the men's restroom and asking people to refer as 'him'. What you can do about it are two things: not ask about it and just leave it be or be honest and talk to him directly, being polite, saying that you noticed the situation and if he would like to be referred as 'she', etcetera. Some people take a long time until they figure out their identity, so maybe he also needs some space to define it until sharing it with the world. Think that he is very brave for doing that because he is figuring out his own label and with a society that judges all the time, it is hard to take these small steps.