Less likely. Less likely caused by a fall, but certainly worsened by a fall. Facet arthtitis is usually caused by disc space narrowing, allowing the facets joints inappropriate movement. This causes secondary osteoarthritis of the facet joints. Any fall would invovle multiple levels, so I believe unlikely.
Not inmediately. There can be situations of an injury to rhe facet that can lead to post traumatic arthritis to set in causing pain symptoma related to the facet joints. This takes time to occur.