Perhaps your backache is posture-induced? Do a complete check-up with a spine expert to establish the cause. If nothing serious is indicated, you may right away start with these poses for relief. If, however, there is a problem, then you must follow the prescription (traction, physiotherapy, etc) as the spine expert suggests. About sleeping surface: a firm surface is said to be good. It means also a firm mattress..so no need to shift to the floor! About morning stiffness: it is a problem common to most people who may not have a habit of physical exercise. In fact, more than the AC, it may have to do with a late dinner, TV-watching before sleeping (one slumps more while doing so) and lack of a regular physical activity. I suggest you start some simple stretching exercises. You also need to do stretch-breaks throughout the day: often just regular stretching can relieve posture-induced or work-induced backache and stiffness. When our body remains in one position for long, blood circulation to that area is reduced, causing a lot of problems -- blood flow is essential for removing waste, repair, transport of nutrients and oxygen to that area. So stretching takes care of all this.
