Lockable doors and beds didn’t even exist for the vast majority of humanity’s existence and still don’t in many places. Beds aren’t even popular sometimes today. Many Japanese children and their parents sleep on mats on the floor. They also may have interior walls, including the ones of their room, made of rice paper. Are all of those children mentally ill?
To suggest that lacking such things would cause mental illness would be to suggest that only a small segment of humanity has ever been not been mentally ill.
As far as escaping your parents or developing your sexuality, neither a lockable door nor a bed are necessary for those things. You may have to go outside your own home once in a while though.