Regarding question (1): There are many ways to phrase the Markov property. One of the more convenient ones is as follows: if $\Phi$ is a non-negative function measurable with respect to $\mathcal F_{t,\infty} := \sigma\{X_s : s \ge t\}$, then
$$ E[\Phi | \mathcal F_t] = E[\Phi | X_t] \text{ a.s.} $$
Furthermore, by the Doob–Dynkin lemma, the right-hand side is equal a.s. to a Borel function of $X_t$.

Clearly, in our case $\Phi_{t,T} = \exp(-\int_t^T R(Z_s) ds)$ is $\mathcal F_{t,\infty}$-measurable (thanks to continuity of $Z_s$; I assume that $R$ is nice enough — say: locally bounded from below), so clearly
$$ E[\Phi_{t,T} | \mathcal F_t] = F_{t,T}(X_t) \text{ a.s.} $$
for a Borel function $F_{t,T}$.

The true question is whether we can choose $F_{t,T}$ in such a way that $(t,T,x) \mapsto F_{t,T}(x)$ is jointly measurable. I expect that under reasonable assumptions (say: the function $R$ is bounded from below), this is standard, but tiresome; however, I did not attempt to write a detailed proof.

(What is easy is that $\Phi_{t,T}$ is uniformly bounded and almost surely continuous with respect to $t, T$, and hence, by the dominated convergence theorem, $(t,T) \mapsto \Phi_{t,T}$ is a continuous map into $L^1(\Omega, P)$; in particular, $(t,T) \mapsto F_{t,T}(X_t)$ is a continuous map into $L^1(\Omega, P)$.)

The answer to question (2) really depends on your favourite definition of a time-homogeneous Markov process. If we are allowed to use the shift operators $\theta_t$ and expectation $E^x$ for the process started at $x$, then things are pretty clear: we have $\Phi_{t,T} = \Phi_{0,T-t} \circ \theta_t$, and hence
$$ E[\Phi_{t,T} | \mathcal F_t] = E[\Phi_{0,T-t} \circ \theta_t | \mathcal F_t] = G_{T-t}(X_t) , $$
where
$$ G_s(x) = E^x \Phi_{0,s} $$
is a Borel function. Noteworthy, here joint measurability of $(s,x) \mapsto G_s(x)$ presents no difficulties: $s \mapsto \Phi_{0,s}$ is a continuous map into $L^1(\Omega, P)$, and so $s \mapsto G_s(x)$ is continuous for every $x$.

I should also add that there is no simple expression for the conditional expectation of the *Feynman–Kac functional* $\Phi_{t,T}$ in terms of the transition semigroup of the process.