This paper deals with continuous-time Markov decision processes with the unbounded transition rates under the strong average cost criterion. The state and action spaces are Borel spaces, and the costs are allowed to be unbounded from above and from below. Under mild conditions, we first prove that the finite-horizon optimal value function is a solution to the optimality equation for the case of uncountable state spaces and unbounded transition rates, and that there exists an optimal deterministic Markov policy. Then, using the two average optimality inequalities, we show that the set of all strong average optimal policies coincides with the set of all average optimal policies, and thus obtain the existence of strong average optimal policies. Furthermore, employing the technique of the skeleton chains of controlled continuous-time Markov chains and Chapman-Kolmogorov equation, we give a new set of sufficient conditions imposed on the primitive data of the model for the verification of the uniform exponential ergodicity of continuous-time Markov chains governed by stationary policies. Finally, we illustrate our main results with an example.