Q2 should be yes for polarized VHS by a rigidity theorem of Schmid [theorem 7.24, Variations of Hodge structure, Inventiones 1973], which says roughly that a PVHS is determined by the Hodge structure at a fibre plus monodromy. Q1 should follow from this by Deligne's equivalence that you stated. Q3 is OK also. The point is that a group with trivial profinite completion will only have trivial representations into $GL_n(\mathbb{Z})$, since the target is residually finite.

Q2 should be yes for polarized VHS by a rigidity theorem of Schmid [theorem 7.24, Variations of Hodge structure, Inventiones 1973], which says roughly that a PVHS is determined by the Hodge structure at a fibre plus monodromy. Q1 should follow from this by Deligne's equivalence that you stated. Q3 is OK also. The point is that a group with trivial profinite completion will only have trivial representations into $GL_n(\mathbb{Z})$, since the target is residually finite.

Addendum Let me give an elementary argument that a family of princ. polarized abelian varieties over the affine line $\mathbb{C}$ is trivial. Given such a family, we get a holomorphic map $f$ from $\mathbb{C}$ to the Siegel upper half plane. This is a bounded domain, so you can separate points using bounded holomorphic functions. After pulling these back to $\mathbb{C}$, they must all be constant. Therefore $f$ is constant.