Regime-switching processes contain two components: continuous component and discrete component, which can be used to describe a continuous dynamical system in a random environment. Such processes have many different properties other than general diffusion processes, and many more difficulties are needed to be overcome due to the intensive interaction between continuous and discrete components. In this work we give conditions for the existence and uniqueness of invariant measures for state-dependent regime-switching diffusion processes. Also, the strong convergence in the L-1-norm of a numerical approximation is established and its convergence rate is provided. A refined application of Skorokhod's representation of jumping processes plays a substantial role in this work.