Is limelight a common noun

Yes, the Word "limelight" is a common noun, a general Word for a type of theatrical light fixture or a situation in which someone is getting a lot of interest and attention.

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

• West Limelight Street, Boise, ID
• Limelight Hotel, Aspen, CO
• "Limelight", 1952 movie with Charlie Chaplin and Claire Bloom