Very partial answer - I don't think I can comment yet...
I found it helpful to rephrase the statement of the support theorem like this:
Let $$ R\pi_\ast \mathbb Q = \bigoplus _i IC{Z_i}(L_i)[d_i] $$ IC_{Z_i}(L_i)[d_i]$ be the decomposition of the pushforward sheaf. If Z is one of the $Z_i$ appearing in the sum above, then $Z$ is the support of a direct summand of $R^{2d}\pi_\ast \mathbb Q$ (i.e. there is extra stuff appearing in the top cohomology of the fibres). Such a summand only will occur when there are extra irreducible components in the fibres of $\pi$.
In particular, if the fibres of $\pi$ are irreducible, then the only possible support appearing in the direct sum is the whole of V. This means that the pushforward is the IC extension of the local system over the locus where $\pi$ is smooth.
I think this means that in your situation, the answer to (1) is yes (as compactified Jacobians are irreducible for such singularities). I would be interested to find out the answers to (2) and (3).
