This work focuses on a class of stochastic damping Hamiltonian systems with state-dependent switching, where the switching process has a countably infinite state space. First, the existence and uniqueness of a global weak solution is constructed by the martingale approach under very mild conditions. Then, strong Feller property is proved by the killing technique together with the resolvent and transition probability identities. The commonly used continuity assumption for the switching rates in the literature is relaxed to measurability. Finally, some sufficient conditions for exponential ergodicity and large deviations principle for regime-switching damping Hamiltonian systems are provided. Several example on regime-switching van der Pol and (overdamped) Langevin systems are studied in detail for illustration. This is a joint work with Fuke Wu and Chao Zhu.