= Solution
For the natural <filtration> $\mathcal F_t=\sigma(X_s:s\leq t)$, a <stopping time> $T$ takes values in $[0,\infty]$ and satisfies $\{T\leq t\}\in\mathcal F_t$ for every $t$. Its occurrence is decidable from information available by that time.
The <Strong Markov property> says that on $T<\infty$, conditionally on $X_T$, the post-$T$ process is independent of $\mathcal F_T$ and has the law of the original <continuous-time Markov chain> started at $X_T$. In particular $\mathbb E_i[f(X_{T+s})\mid\mathcal F_T]=P_sf(X_T)$ on $T<\infty$. The chain is an <irreducible Markov chain> if every state communicates with every other: for any $i,j$ some time has $\mathbb P_i(X_t=j)>0$, equivalently for finite rates there is a path of positive off-diagonal jump rates between them.
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