Good question! I wished I understood what this conjecture really means, concretely. First, I should say that in my paper with Bhatt on prismatic cohomology, much of the content of these conjectures has been proved, although the connection is not made very clear, and we only work after $p$-adic completion for some prime $p$; it would be possible to go back and get full integral statements by pasting together the rational information and the $p$-adic information.

The main problem in getting something very concrete out of this formalism --- like explicit $q$-periods, or explicit $q$-difference equations deforming the Picard--Fuchs differential equation --- is that it seems very hard to construct explicit vectors in $q$-de Rham cohomology. For getting say the Picard--Fuchs equation, one uses that de Rham cohomology has a Hodge filtration, and one can write down the differential equation satisfied by a vector in the $1$-dimensional subspace. This subspace does not deform to $q$-de Rham cohomology, so one gets no canonical $q$-difference equation. On the other hand, weight filtrations should still be defined, and maybe one can use those in some cases to get interesting explicit $q$-difference equations.

Still, Question 1 seems to be largely answered in this paper by Ryotaro Shirai. If I understand it right, he restricts to the ordinary part of the moduli space of elliptic curves, in which case there is a canonical "ordinary part" of the $q$-de Rham cohomology, giving a canonical $1$-dimensional subspace, for which one can write down the $q$-difference equation. Shirai then shows that this is indeed a hypergeometric $q$-difference equation.

For Question 2: This isomorphism is only canonical after going to one of Fontaine's period rings like $A_{\mathrm{inf}}$. Doing this, one essentially ends up with the usual $p$-adic periods, living in Fontaine's field $B_{\mathrm{dR}}$ (or subrings thereof).