It's perfectly normal to have questions about your therapist. Some questions are easier to ask your therapist, such as "Where did you obtain your degree?" or "How long have you been practicing?" Other questions, however, can feel more difficult to ask, such as "Have you ever struggled with depression?" or "Have you gone to therapy?"
Guess what: you are allowed to ask! Your therapist may choose not to answer, but it is not offensive to ask. Generally, it also isn't crossing any boundaries.* And, regardless of whether your therapist chooses to answer or not, why you were wondering can be an important topic to discuss in the session.
If your therapist does choose to answer, they're engaging in what is called "self-disclosure." The extent to which a therapist uses self-disclosure depends on a number of things, including their level of comfort sharing personal information and their theoretical perspective. A good therapist will also consider how the self-disclosure will affect the specific client they're working with.
For example, I work from a feminist perspective. Feminist therapy encourages self-disclosure as a way to demystify and equalize the therapeutic relationship. Also, I'm also generally pretty comfortable sharing about myself. So, although there are certainly reasons why I might not use self-disclosure in a particular situation or with a particular client, there are times when I will.
I might use self-disclosure as a way to normalize a client's symptoms or concerns. I might say something like, "You know, I think what you're experiencing is more normal than you think. In fact, I actually felt that same way when I was applying for graduate school." Or, I might use self-disclosure as a way to encourage a client to try something new: "When I don't want to get out of bed, I visualize myself saying no to the bed and the tricks it uses to keep me there. It can feel silly at first, but it really helps me feel empowered. What do you think it would be like for you?"
*There are exceptions. One example of a question that would cross boundaries is "Will you go on a date with me?" That said, it is not crossing boundaries to tell your therapist that you have been thinking about going on a date with them and aren't sure what to do about it or what it means. Exploring these feelings -- why they have arisen and how you can manage them in order to remain in therapy -- can be an important therapeutic task (of course, your therapist should make it clear that dating a client is a major ethical violation that they can't and won't engage in).