If you are changing your domain name, perhaps because you have changed the product that you are promoting, but you have an existing customer base at a previous domain name, would you use a redirect? If so, how long for?

Probably "forever", and I would use a 301 redirect, so that the first site's link popularity, would not be totally lost. $10 per year or so, to keep a domain name, is peanuts for a business.