Is the name Kate a proper noun?

Yes, Kate is a proper noun.

Proper nouns (also called proper names) are nouns representing unique entities (such as London, Venusor Kate), as distinguished from common nouns which describe a class of entities (such as city, planet or person). Proper nouns are not normally preceded by an article (a, the) or other limiting modifier (such as any or some), and are used to denote a particular person, place, or thing without regard to any descriptive meaning the Word or phrase may have.