The mean first exit (passage) time characterizes the average time of a stochastic process never leaving a fixed region in the state space, while the escape probability describes the likelihood of a transition from one region to another for a stochastic system driven by discontinuous (with jumps) Lévy motion. This paper discusses the two deterministic quantities, mean first exit time and escape probability, for the anomalous processes having the tempered Lévy stable waiting times with the tempering index $λ>0$ and the stability index $0<α\le 1$; as for the distribution of jump lengths or the type of the noises driving the system, two cases are considered, i.e., Gaussian white noise and non-Gaussian (tempered) $β$-stable ($0<β<2$) Lévy noise. Firstly, we derive the nonlocal elliptic partial differential equations (PDEs) governing the mean first exit time and escape probability. Based on the derived PDEs, it is observed that the mean first exit time depends strongly on the domain size and the values of $α$, $β$ and $λ$; when $λ$ is close to zero, the mean first exit time tends to $\infty$. In particular, we also find an interesting result that the escape probability of a particle with (tempered) power-law jumping length distribution has no relation with the distribution of waiting times for the model considered in this paper. For the solutions of the derived PDEs, the boundary layer phenomena are observed, which inspires the motivation for developing the boundary layer theory for nonlocal PDEs.