The classical double copy maps exact solutions of general relativity to exact solutions of U(1) Yang-Mills theory and suggests a hitherto unknown connection between gravity and gauge theory. In this thesis I study three problems using the Kerr-Schild and Weyl formulations of the classical double copy. Using the Kerr-Schild double copy, I analyze the single copy of a rotating nonsingular black hole and analyze its horizon structure to probe the relationship between the presence of horizons on the gravity side and the single copy field on the gauge theory side. In the second problem I describe the mapping between the surface gravity of static spherically symmetric black holes and the force on a test particle due to the single copy field of the black hole. I also describe potential routes to extending this map to rotating black holes. Finally, inspired by the extended Weyl double copy for spacetimes possessing sources, I reinterpret the single copy of the Taub- NUT metric as being comprised of two terms each being sourced by a separate parameter (the mass and the NUT charge).