Yes, Black Golden Retrievers are rare as they are genetic mutations of the traditional Golden Retrievers. Black Golden Retrievers are the result of crossbreeding of multiple dog breeds for several generations to get the desired black fur.
Yes, Black Golden Retrievers do exist. They are genetically linked to Golden Retrievers along with other breeds like Flat-Coated Retrievers (or Black Retriever dogs), Black Labrador Retrievers, Black German Shepherds, and Black Goldendoodles.
Black Retrievers/Flat-Coated Retrievers, resemble Golden Retrievers, except that the Black Retrievers are black in color and have less wavy fur coats. Black Golden Retrievers look more like Golden Retrievers (except for the color, of course) with their feathery coats and body composition.