Is it better to heat or ice spinal stenosis pain?

Tough diagnosis. Work with a good physical therapist. Find musculoskeletal acupuncture such as trigger points (don't need injections, just 'dry needling'). A good hands-on osteopathic physician who does omt (omm) is key for maintaining function in a chronic situation. Finally since this is really arthritis in the spine, and has an inflammatory component, eat an anti-inflammatory diet and herbs (find online).
Both. Some people get more relief from heat or ice. Others alternate heat and ice. The most important thing you can do is exercise to strengthen your paraspinal muscles.
Varies. It varies somewhat. In general, moist heat helps with pain related to spasm where as cold can help "numb" pain coming from superficial structures. Either is ok to try, and you can use which works better for you or alternate between the two.