Let $M$ be a model of $\sf ZFC$ in which $\kappa$ is a measurable cardinal, and $\cal U$ is a normal measure on $\kappa$. We can define the Prikry forcing (the most simple one) as the poset: $$\Bbb P=\left\{(p,A)\mid p\in[\kappa]^{<\omega}, A\in\mathcal U, \max p<\min A\right\}.$$
We also define the order, $(q,B)$ is stronger than $(p,A)$ if $q$ is an end-extension of $p$, $B\subseteq A$ and $q\setminus p\subseteq A$. The generic $G$ adds an $\omega$-sequence, $x\_G=\bigcup\{q\mid\exists A:(q,A)\in G\}$ which we call a *Prikry sequence*. We can show that $x\subseteq^\* A$, for all $A\in\cal U$, that is $x\setminus A$ is finite.


In the other direction, if $M$ is a model of $\sf ZFC$ in which $\kappa$ is measurable, and $M\subseteq V$, such that in $V$ we have some $x\in[\kappa]^\omega$ such that for some $\cal U$ in $M$ which is a normal measure on $\kappa$, $x\subseteq^\*A$ for all $A\in\cal U$, then $x=x\_G$ for some generic filter $G$ over the Prikry forcing defined from $\cal U$.


One conclusion from this last statement is that if $x$ is a Prirky sequence for $\kappa$, and $y\subseteq x$ is infinite then $y$ is also a Prikry sequence. This raises the following question.



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> Suppose that $x$ is a Prikry sequence for $\kappa$, and $y\subseteq x$ is infinite. Is $x$ generic over $y$? More generally, if $y,w\subseteq x$ are disjoint infinite subsets, are they pairwise generic? What about the weaker condition $y\cap w$ being finite?
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Is Prikry forcing minimal?
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 $x=G\_C$ is $M[y]$-generic for $P/y$ and $M[G\_C]$ is also $M[y][G\_C]$. 


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>


