Let $\alpha > 0$ be any ordinal. Consider the forcing $Add(\omega,\alpha) * Add(\omega_1,1)$. Let $B$ be its boolean completion. Let $\dot{X}$ be the canonical $B$-name for the generic subset of $\omega_1$ added by the second iterand. By forcing folklore, there is a complete subalgebra $A \subseteq B$ that adds $X$, completely generated by the boolean values $|| \check{\beta} \in \dot{X} ||$ (see Jech p. 247). One can check that for each $\beta < \omega_1$, $|| \check{\beta} \in \dot{X} || = (1,\check{ \lbrace (\beta,1) \rbrace } ) $.

Questions:

(1) What is the nature of $A$ in relation to $B$? Although $A$ adds all the reals of $B$, I think I can show that it is always a proper subset of $B$ (even for $\alpha = 1$). This leads to the second question.

(2) If $G$ is generic for $A$, what is the nature of the quotient algebra $B/G$? One can show that it is $(\omega,\infty)$-distributive. In the case where $\alpha$ is countable, it is atomic. Is it nonatomic for uncountable $\alpha$? Is it strategically closed?

(3) If $G$ is generic for $A$, does $V[G]$ always contain *some* $Add(\omega,\alpha)$ generic, like in the case $\alpha$ is countable?

Edit: I thank J.D. Hamkins for his answer, but I already knew about the case $\alpha < \omega_1$, as stated in the original post. I would really like some insight on the case $\alpha \geq \omega_1$.

**NEW RELATED QUESTION**

Consider instead $Col(\mu,<\kappa)*Add(\kappa,1)$, where $\mu< \kappa$ are regular. Let $B$ be the completion, and let $A$ be the complete subalgebra generated by the canonical name for the Cohen subset of $\kappa$ added by the second part. Let $H \subseteq A$ be generic.

**Does $B/H$ have a $\mu$-closed dense subset in $V[H]$?** Any proof or refutation would be great, or a discussion of other related structural aspects of $A,B$, and $B/H$. The properties of this forcing in relation to large cardinals are important to my research, so all nontrivial information is welcome.