On Wednesday, November 9, 2016 at 6:36:30 PM UTC-8, wrote:
I just bought a nice 6'x9' or so roll of carpet at Home Depot for around $18. Very short pile and no backing.

Tom

Automotive type carpeting generally sold at auto re upholstery stores (not floor covering) are light weight and pliable. Light weight is the magic here, makes no difference cut pile or loop, generally cut pile will work better as it won't unravel.
As far as glues - contact cement is what you want but be careful not to buy the "green" labeled ones as they simply don't work. The old fashioned stuff unfortunately is only sold in small cans, makes it more expensive but it is still worth it, make sure you know how to use it or better yet get advise from someone who does- you could end up with sagging messy carpet all over the place and very sticky fingers otherwise.
Just so happens to be in the business for more decades than I care to admit....