Suppose that I want to force to add a "single" new subset of $\omega$ and not much else. For example, consider the Cohen forcing consisting of finite partial functions from $\omega$ to 2. The forcing I am interested in is different (in fact not CCC, but Proper), but still the generic set codes a subset of $\omega$.
Question 1: Suppose that $G$ is $P$-generic over $M$, and there is at least one $r\in(\mathscr{P}(\omega)\cap M[G]) \setminus M$. Is there a condition $(*)$ such that
$P$ satisfies $(*)$ if and only if $\forall t\in M[G]\cap \mathscr{P}(\omega) \exists a, b\in M\cap \mathscr{P}(\omega) [ t = (a \cap r) \cup (b\setminus r)] $ ?
Question 2 (Iterability): Consider a model $M_\delta$ resulting from an iteration of forcings $P_\alpha : \alpha<\delta$. Say that each $P_\alpha$ adds a real $r_\alpha$. Then $ \{r_\alpha : \alpha<\delta\} \subseteq \mathscr{P}(\omega)\cap M_\delta$. Is there a model $M_\delta'\subseteq M_\delta$ such that the reals of $M_\delta'$ are just $(\mathscr{P}(\omega)\cap M)\cup \{ (a\cap r_\alpha)\cup (b\setminus r_\beta) : a,b\in \mathscr{P}(\omega)\cap M; \ \alpha, \beta<\delta\} $ ?
