I don't think it's inappropriate at face value, but I'd never expect a full response if that makes sense. It has helped me to look at my relationship with my therapist just like I look at my relationship with my students just in reverse. When my students ask me questions, some of them are completely ridiculous and inappropriate, but I can honestly say I don't take them personally.. the 4 times I've been asked if I was pregnant for example LOL Some they ask are more innocent but still none of their business... Why don't you have children? Do you and your siblings all have the same father? etc etc.. and sometimes I answer, sometimes i deflect, depending on the student and the circumstance and why they are asking.

I spoke to one of my classes about my father dying because it was poignant to the conversation... but I would never go into that detail just on a random conversation.

I say ask what you want.. and then just trust your t to not make you feel bad/shame/guilt about it... Trust that your T will support you through it.