For Question 1, the answer is no. In fact, there are pseudo-Anosov maps $\psi$ that preserve a handlebody $H_1$ (the existence of such a map follows from work of Masur-Minsky, although probably appears earlier). $H_1\cup_{\psi^n} H_2 = H_1\cup_S H_2$, so if $H_1\cup_S H_2$ is not hyperbolic, then neither will $M_{\psi^n}$.

For Question 2, the answer is a bit more complicated. Namazi-Souto prove if $\psi$ is "generic", which means that the stable lamination $\lambda_+$ of $\psi$ is not a limit of meridians of $H_2$, and the unstable lamination $\lambda_-$ of $\psi$ is not a limit of meridians of $H_1$, then $M_{\psi^n}=H_1\cup_{\psi^n} H_2$ is hyperbolic for large $n$. Moreover, for any $\epsilon >0$ and large enough $n$, there are metrics on $M_{\psi^n}$ with curvatures pinched between $-1-\epsilon$ and $-1+\epsilon$, with volume growing linear with $n$. This implies that the hyperbolic volume of $M_{\psi^n}$ grows linearly as well, either by applying volume comparison theorems of Besson-Courtois-Gallot, or by an unpublished preprint of Tian.

If the map $\psi$ is not generic, (say $\lambda_+$ is a limit of meridians of $H_2$), then Biringer-Johnson-Minsky prove that a power of $\psi$ extends over a compression body inside of $H_2$. This does not necessarily imply that the manifolds $M_{\psi^n}$ is not hyperbolic, but if they are I'm not sure how fast the volume grows; I suspect it would still grow linearly if some power does not extend entirely over $H_1$ or $H_2$ (like in the answer to Question 1).