I don't think so. I would use the question only to ask someone if they still have their room, to know whether they have given it up or not. I would use 'Do you have a room?' and the same with the answers- we have a room/we have our own room/ we have a room to ourselves.
BTW, I would use apartment/flat unless you really mean a single room, and a one-room flat is a studio. I know that many Japanese speakers don't distinguish between the two, but it sounds strange to me- my Japanese partner uses room for our apartment/flat and it sounds really weird to me.
The situation I have in mind is whether a child has his own room in his house or not. In Japan, many children have to share their rooms with their brothers or sisters especially when they are small. That's because Japanese houses usually don't have so many rooms that every child in the family has his own room.
Anyway, from the explanation you two gave me, I don't think we can say "Do you have your room?". And rather we should say, "Do you have your own room?" or something like that.