This work focuses on a class of stochastic Hamiltonian type jump diffusion systems with state-dependent switching, in which the switching component has countably infinite many states. First,the existence and uniqueness of the underlying processes are obtained with the aid of successive construction methods. Then, the Feller property is established by coupling methods. Furthermore, the strong Feller property is proved by introducing some auxiliary processes and making use of appropriate Radon-Nikodym derivatives. Finally, on the basis of the above results, the exponential ergodicity is obtained under the Foster-Lyapunov drift condition.