What are the most popular ingredient used in the making of gluten-free beers? Do they have a good shelve life or not?
Corn, rice, sorghum, buckwheat, millet and quinoa are popular. Each has their own special trick to get it to convert the starches to sugars.
In Africa sorghum lagers are very popular, and they have the same shelf life of "normal pale" lagers. aka 3-6 months. I do not know if gluten-free affects the shelf life.
Here is a BYO article to provide more information on gluten free brewing.
IIRC, Lakefront New Grist is made from sorghum. I can't comment on shelf life.