14 months ago I fell on an icy sidewalk. A day later, pain began in the lumbar region of my back. Can you break your back with no neuro symptoms?

Yes. If you did fracture one of your vertebrae it would be healed by now. If you still have significant pain you should be evaluated for another cause for your back pain.
Yes. Neuro symptos come from nerve ptessure there are many ways to fracture you spine wthat dont result in nerve pressure most acute back pain after a fall is a muscle strain from muscluar contraction involved in the righting reflex see yoiro dovctor get a physical exam rule out fracture with xray and treat the muscle pain. If the back muscles is tender to the touch it is probaly a muscle spasm.
Yes. You can but if you broke your back you probably wouldn't be able to walk very well. It may be a muscle sprain. However, you should get it checked out soon anyways.