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Can the magnetic and electric potentials ($A$ and $φ$) be fixed to zero, or at least some constant value in a region of space? Naively, I'd think something like this might work (a hollow conducting sphere connected to a voltage source connected to ground, would the potentials inside the sphere be constant?):
Yes, in a conductor you have that inside the electric field is zero so $ \boldsymbol{E}= - \nabla \phi=\boldsymbol{0} \rightarrow \phi= \text{constant} $
