I am trying to understand the proof of the following lemma from the paper Can the fundamental group of a space be the rationals? by Saharon Shelah.
Let $\mathcal{E}$ be an analytic equivalence relation on $\mathcal{P}(\mathbb{N})$ such that when $n\notin B$ and $A=B\cup\{n\}$, $A$ and $B$ are not $\mathcal{E}$-equivalent. Then there is a perfect subset of $\mathcal{P}(\mathbb{N})$ of pairwise nonequivelent $A\subseteq\mathbb{N}$.
I am aware that Pawlikowski gave a neat proof of this using just basic descriptive set theory, but I want to understand the logic of this proof. The proof goes as follows (some phrases slightly altered).
Let $M$ be a countable elementary substructure of $(H(\mathfrak{c}^+),\mathcal{E})$ to which the real parameter in the definition of $\mathcal{E}$ belongs. Now if $\langle A_1,A_2\rangle$ is a pair of subsets of $\mathbb{N}$ which is Cohen generic over $M$, then they are $\mathcal{E}$-equivalent iff they are $\mathcal{E}$-equivalent in $M[A_1,A_2]$, by the absoluteness criterions. Now we show that they cannot be $\mathcal{E}$-equivalent in $M[A_1,A_2]$. Otherwise some finite information forces this, so for some $n$, if $\langle A'_1,A_2\rangle$ is Cohen generic over $M$ and $A_1\cap\{0,1,...,n\}=A_1'\cap\{0,1,...,n\}$, then $A_1',A_2$ are $\mathcal{E}$-equivalent in $M[A'_1,A_2]$. Let $A_1'$ differ from $A_1$ at $n+1$ only (if $n+1\in A_1$ then $n+1\notin A_1'$, and vice versa). Clearly $\langle A'_1,A_2\rangle$ is also Cohen generic, so they are $\mathcal{E}$-equivalent in $M[A'_1,A_2]$. By the above absoluteness we really have $A'_1\sim A_2$ and $A_1\sim A_2$, so $A'_1\sim A_1$, contradicting the property of $\mathcal{E}$.
I want to ask questions about pretty much every line, but here are the points that I feel most confused about:
What's the definition of $N[G]$ when $N$ is non-transitive? From Hamkins' answer to this question it seems essentially we have to take the collapse of $N$ and then do forcing, so $N$ is not a subset of $N[G]$. Is my understanding correct?
If $N$ contains the "real parameter defining $\mathcal{E}$", isn't $\mathcal{E}$ already an element of $N$, since $\mathcal{E}$ is definable in $H(\mathfrak{c}^+)$ from parameter and $N$ is an elementary substructure (at least that should be the case for the particular $\mathcal{E}$ in the paper)? What's the meaning of $(H(\mathfrak{c}^+),\mathcal{E})$ then?
What does "absoluteness criterions" refer to? Is it Shoenfield's Absoluteness? Does that apply to model without power set (plus $\mathcal{P}(\mathbb{N})$ exists)?
Where is it used that $\mathcal{E}$ is analytic?
Thank you in advance for your answer.