Implications of the concepts of combination/expansion of an information structure
Answer to Question 1: Yes. To see this, note that for every combination of elements of $T^1\times T^2$ in the support of $\pi^*$ you get a, possibly different, posterior. To show $\pi^*$ is at least as informative as $\pi^1$ you need these posteriors to be a mean-preserving spread the posteriors induced by $\pi^1$ (there are actually several characterizations of Blackwell's partial order, I'm thinking there is a more elegant proof, but this is the one I can think of right now). You can easily show this is the case if you average up the posteriors induced by $S^*$, over the realizations of $T^2$.

Answer to Question 2: Yes. For any signal $S$, you can find a "trivial" expansion by combining it with $\underline \pi$, so the resulting signal $S^*$ (which is informationally equivalent to $S$) is at least as informative than $\underline S$.

In that sense, Bayesian decision makers can never have less information than their prior.