 Solution

For the natural filtration ${\mathcal{F}}_t=\sigma (X_s:s\le t)$, a stopping time $T$ takes values in $[0,\infty ]$ and satisfies $\{T\le t\}\in {\mathcal{F}}_t$ 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 ${\mathbb{E}}_i[f(X_{T+s})\mid {\mathcal{F}}_T]=P_{s}f(X_T)$ 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.