Will a previous lumbar fracture cause spinal stenosis in the future?

Sometimes. A prior lumbar fracture can cause spinal stenosis if some of the fracture fragments get pushed into the spinal canal where the nerves are. Usually the fracture will cause stenosis acutely around the time of the fracture and not usually in the distant future. However, some fractures can result in delayed stenosis and should be evaluated by a physician.
Possible. Sometimes a spine fracture can compromise the size of the spinal canal at the time of injury or lead to accelerated degenerative changes that can occur over time leading to the stenosis or even a late instability can develop creating stenosis. Sometimes, a fracture can lead to a deformity like a kyphosis which can stretch the spinal cord making it injured or more suscepitble to any compression.
Sometimes. It varies, but yes it potentially can. If the fracture displaced it can cause stenosis, or if arthritis sets in stenosis can develop as well.