We study finite horizon optimal stopping problems for continuous-time Feller-Markov processes. The functional depends on time, state, and external parameters and may exhibit discontinuities with respect to the time variable. Both left-and right-hand discontinuities are considered. We investigate the dependence of the value function on the parameters, on the initial state of the process, and on the stopping horizon. We construct epsilon-optimal stopping times and provide conditions under which an optimal stopping time exists. We demonstrate how to approximate this optimal stopping time by solutions to discrete-time problems. Our results are applied to the study of impulse control problems with finite time horizon, decision lag, and execution delay.