Am I allowed to ask my therapist questions about their life?
As a client, you are allowed to ask your therapist just about anything. And, it is possible that the therapist will not or cannot answer the question for a variety of reasons.
Some counselors believe strongly in being a "blank screen" or "mirror" in therapy. They believe they can be of most help to their clients if their clients know as little about them as possible. How could knowing something about your counselor be harmful? For example, if I tell my clients that I have dogs and they have an intense fear or disgust of dogs, they may be less forthcoming about their true feelings in order to spare mine.
Other counselors disclose quite a bit about their own experiences in treatment. For example, it is not uncommon for some drug and alcohol counselors to identify themselves as "recovering." This isn't exclusive to drug and alcohol counselors either. Famously, Marsha Linehan, disclosed her own struggles with mental illness in this NY Times article: http://www.nytimes.com/2011/06/23/health/23lives.html?pagewanted=all&_r=0.
Most counselors are probably somewhere in between these two positions. The only global advice I have is that you should ask your counselor how comfortable they are with personal questions. They shouldn't make you feel bad about being curious. Even if they can't answer, knowing the questions you have helps us understand you better and may help us provide you with better therapy. Don't be shy!
When we can't answer a question, you may wonder, "what secrets does this person have?" Often if we refuse it's because we want to protect our clients from learning something that might set their therapy back. But it's usually not because of a big dark secret. For example, if I am seeing a couple that is fighting about whether their kids get to eat sugar, I will probably not share with them that I let my child have a bite of pastry last night, because they might assume I'm taking a side. Our job is to help your live YOUR best life - not to impose our values and ideas upon you.
So please, ask away! Just remember that your time with us is supposed to be about you so if we can't answer a question about ourselves, it's probably because we want to keep the focus where it belongs.