I have the answers to my two questions. (I've actually had them for a while; apologies for the delay in posting.) I will give them in reverse order:

The answer to Q2 is *yes*; the structure of the proof is exactly as I outlined in the update. Details can be found in section 5 (in particular, Corollary 100) of my notes http://wwwf.imperial.ac.uk/~jmn07/Ergodic_Theory_for_Semigroups_of_Markov_Kernels.pdf.

The answer to Q1 is also *yes*: Since $f$ is bounded, it is sufficient just to consider the limit as $T$ tends to $\infty$ in the integers. By the positive answer to the question https://mathoverflow.net/questions/181640/is-it-true-that-all-stationary-measurable-stochastic-processes-are-measurably-s, the discrete-time stochastic process $\left(\int_n^{n+1} f(X_t(\cdot)) \, dt \right)_{n \geq 0}$ is stationary, and therefore Birkhoff's ergodic theorem (applied to the shift map on $\mathbb{R}^{\mathbb{N}_0}$) gives the desired convergence.