There is a theorem of Gitik, Kanovei and Koepke that characterises the degrees of constructibility in $M[G]$ where $G$ is prikry generic over the model $M$: they are isomorphic to the $P(\omega)/Fin$ of $M[G]$.
That is they show (letting $M, G$ be as above with $x=G_C$ the actual prikry sequence):
$$∀Z ∈ M[x] ∃y ⊆ x, y ∈ M[x] \wedge M[Z] = M[y].$$
They identify a $\kappa^+$-c.c quotient p.o. $P/y$ needed to see that if $y\subseteq x$ is an infinite subset of the Prikry generic $x$, then $G$$x=G_C$ is $M[y]$-generic for $P/y$ and $M[G]$$M[G_C]$ is also $M[y][G]$$M[y][G_C]$.
This in short, I think answers your questions; (at least the answer to Q1 is "Yes". I am not quite sure what you mean by "pairwise generic" for the case of $y,w$ disjoint - say the Evens and Odds in the sequence $x$ - but anyway, yes in the above sense: $w$ is generic for a quotient forcing over $M[y]$). (This includes your `weaker version'.) I don't know if the result is published, but there are slides with quite a bit of detail of Koepke at:
http://www.math.uni-bonn.de/people/koepke/Talks/Submodels_of_Prikry_generic_extensions.pdf