Regarding your question on Hodge structures on $\pi_1(X)$, what we know is that:
for any topological space $X$ there exists a complex $A^*(X)$ equipped with a structure of commutative differential graded algebra that computes the singular cohomology of $X$: $$H^*(A^*(X))\cong H^*(X,\mathbb{Q})$$
in rational homotopy theory "à la Sullivan" this complex determines when $X$ is nilpotent and of finite type the rational homtopy type of $X$ i.e. we can compute the rational homotopy groups of $X$ from $A^*(X)$. Otherwise in the non-nilpotent case we only get information on the lower central series of $\pi_1(X)$.
when $X$ is complex algebraic variety (possibly singular and non-proper) we can put a mixed Hodge structure on $A^*(X)$ that induces the MHS on $H^*(X,\mathbb{Q})$ as described by Deligne in his foundational papers.
we can extract from this Hodge structure on $A^*(X)$ a MHS on the lower central series of $\pi_1(X)$ and when $X$ is nilpotent of finite type we get a MHS on the higher homtopy groups of $X$.
in general the MHS on $H^*(X,\mathbb{Q})$ does not determine the MHS on $\pi_1(X)$, you need to go to the model $A^*(X)$ together with its MHS. However when $X$ is said formal that is when $A^*(X)$ is quasi-isomorphic to $H^*(X,\mathbb{Q})$ as a commutative differential graded algebra, this MHS is determined by the MHS on $H^*(X,\mathbb{Q})$.
Examples of formal complex algebraic varieties are given by smooth and proper ones.