Is carpet a common noun

Yes, the noun carpet is a common noun, a Word for any carpet of any kind, anywhere.

A proper noun is the name of a specific person, place, thing, or a title; for example:

• Mr. Carpet's Floor Shop, Decorah, IA
• Carpet Street, Enfield, CT or Carpet Street, Prichard, AL
• Red Carpet Inn, Orlando, FL
• "The Magic Carpet", 1951 movie with Lucille Ball
• "Carpet Ride", a novel by Norm Brown