Suppose I have a smooth projective family of varieties of varieties over $\mathcal M_g$ - i.e. a universal functor, commuting with deformations, from curves to smooth projective varieties. Can I compute its cohomology directly from the cohomology of the curves?

In some cases, like the functor associating a curve $C$ to the smooth projective variety $\operatorname{Sym}^n C$, I certainly can. Can I in general?

I think the right way to formalize this is to consider your favorite substitute for the conjectural category of relative motives - e.g. either $\ell$-adic sheaves or variations of Hodge structures. Consider the Tannakian category of relative motives on the stack $\mathcal M_g$ and take its Tannakian fundamental group $\pi_1^{mot} (\mathcal M_g)$. Then consider the interesting part, which I'm going to take to be the maximal reductive quotient of the identity component of $\pi_1^{mot} (\mathcal M_g)$. One should also take the kernel of the natural map tot he motivic Galois group of the base field.

Is this equal to $SP_{2g}$, the part coming from the universal family of curves, or is it larger?

One way to answer this would be to exhibit a smooth projective family over $\mathcal M_g$ with weird cohomology.

For $\mathcal A_g$, $g \geq 2$, I think you can show that the identity component of the motivic fundamental group (defined in an $\ell$-adic way) is exactly $SP_{2g}$ using the congruence subgroup property.

I'm also interested in the case where we restrict to motives with good reduction at every place.