Gauss Manin connection for the study of variation of arithmetic periods

I've been interested in the Riemann-Hilbert correspondence.
One of the application in number theory seems to be the following. Let us consider a family $\pi: X \to S$ (smooth proper) of varieties over $k \subseteq \mathbb{C}$. I can extract from different ways a connection, the Gauss-Manin connection $\nabla_{GM}$ from this family (the first one of them is by using the RH correspondence).
Now, one considers a function coming from the comparison isomorphism :

$\displaystyle F(s) = \int_{\sigma_s} \omega_s$,
for locally constant Betti classes $\sigma_s$, and de Rham classes $\omega_s$.

I have seen in lecture notes by S. Bloch the following formula :

(*) $\displaystyle d_S(F(s)) = \int_{\sigma_s} \nabla_{GM}(\omega_s)$,
and by the fact that de Rham cohomology vector spaces are finite dimensional, I can find a linear relation between $\omega_s, \nabla_{GM}(\omega_s), \dots$ that gives me the Picard-Fuchs equations, to describe the variation of the period map $F$.

My questions are the followings:

1. Have I made any major mistake ?
2. How does (*) holds ?
3. What field(s) $k$ is interesting to consider, and how does this matter ?
4. How is this related to Hodge theory ?
5. What are the classical references for these types of questions ?
6. Can the $\mathcal{D}$-modules theory be a better way to express this ?
7. Does a $p$-adic analogous exists ?

NB : I'm not very familiar with periods theory or Hodge theory in general.