Families of abelian varieties on the line (or more generally simply connected varieties)

Here is a partial answer to questions 1/3. Let be given an abelian scheme $\mathscr A/V$. Let us assume that it carries a principal polarization; in principle, we may reduce to this case by an appropriate isogeny but I do not know whether there is a reference in the litterature. For every integer $n\geq 1$, the $n$-torsion subscheme $\mathscr A_n$ is finite étale over $V$, hence is trivial because $\pi_1(V)=0$. The generic fiber of $\mathscr A/V$, an abelian variety over the field $\mathbf C(V)$ which admits level structures of all level. When $V$ is the affine line, this contradicts a theorem of Alan M. Nadel (“The Nonexistence of Certain Level Structures on Abelian Varieties over Complex Function Fields”, Annals of Mathematics, Vol. 129, No. 1 (Jan., 1989), pp. 161-178).

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.

Here's a bit of a long comment which I hope will help the OP. I also answer some of the questions that arose in the comments.

Firstly, any smooth complex algebraic quasi-projective variety carrying an immersive period map is known to be "hyperbolic" in the sense that

i) it is Brody hyperbolic,

ii) all its subvarieties are of log-general type, and

iii) the fundamental group of $X$ is infinite (and more...).

The hardest of these to show is ii) which follows from a theorem of Kang Zuo; see Theorem 0.1 in https://www.researchgate.net/publication/252994947_On_the_negativity_of_kernels_of_Kodaira-Spencer_maps_on_Hodge_bundles_and_applications

or the more general

Theorem 0.3 in Brunebarbe's http://sma.epfl.ch/~brunebar/Articles/Symmetric%20differentials%20and%20variations%20of%20Hodge%20structures.pdf

These results are used in Lemma 6.3 of http://arxiv.org/pdf/1505.02249v1.pdf . The statement of Lemma 6.3 is a consequence of the statement in Brunebarbe as any PVHS coming from geometry is unipotent (up to finite etale base-change of the base). The result of Zuo is also used in the proof of Proposition 3.1 in Abromovich--Varilly-Alvarado's http://arxiv.org/pdf/1601.02483v2.pdf . I mention these statements as they provide a more direct link to your questions.

To prove iii) you can use the argument given by Arapura or use Theorem 3.1 in Voisin's second book on Hodge theory (which is the same argument as in Arapura's answer).

The above results of Zuo (and Brunebarbe) tell us that all horizontal (smooth) algebraic subvarieties of period domains of PVHS's are hyperbolic. This is I think also the best one could hope for: the period domains of PVHS's are not in general "hyperbolic".

Finally, the argument in Arapura's answer is applied to the stack $\mathcal {A}_{g,1}^{\textrm{an}}$ or, if stackiness bothers you, to the fine moduli space of $g$-dimensional ppav's with full level $3$-structure. It is, in my opinion, unnatural to consider the coarse moduli space (which is not hyperbolic, as it contains copies of the $j$-line) when studying hyperbolicity of parameter spaces of ppav's (or smooth proper curves of genus at least two, or polarized K3 surfaces, polarized Calabi-Yau manifolds, etc.). Note that Siegel upper half-space $\mathbb H_g$ is the universal cover of the stack $\mathcal {A}_{g,1}$. The holomorphicity of a lift $\mathbb C\to \mathbb H_g$ for a holomorphic map $\mathbb C\to \mathcal A_{g,1}^{\textrm{an}}$ is therefore immediate.

I can't give you an intuitive reason as to why $\mathcal A_{g,1, \overline{\mathbb F_p}}$ is no longer hyperbolic when $g\geq 2$. If $g=2$ the non-hyperbolicity comes from the supersingular locus. Does that help?