Yes, first build a well-founded model of Zermelo with Urelements, where the $a$'s and $b$'s are the urelements, and then quotient out the intended relationship in the resulting model

$\kappa$ has the property $`` \exists \alpha < \kappa$ such that $V_{\alpha} \equiv_{\text{SOL}} V_{\kappa}"$ because $M$ thinks $j(\kappa)$ has that property.

You’re right about the typo. I’m not assuming ill-foundedness in $V,$ I’m defining $b_p$ in terms of how they’re intended to be interpreted in $M.$ A formal definition would be by a quotient structure, same as how Quine atoms are included in the model.

Yes measurables work, because for $j: V \rightarrow M$ with critical point $\kappa,$ $V_{\kappa+1}^M$ and $V_{j(\kappa)+1}^M$ have the same theory.

If there’s $\mathfrak{c}^+$ inaccessibles, some $V_{\kappa}$ has the same theory as a smaller one. Now treat that as your $V.$

This is very similar to your last question, so I'm not going to write out a full proof, but this is also equivalent to "there is a function choosing an enumeration of each countable ordinal." If you have a countable closed set of rank $\alpha,$ you can canonically enumerate $\alpha$ by sending a basic open set $U$ to $\beta$ if there is a unique point of maximal rank in $U,$ and that rank is $\beta.$

By the way, you might be interested to know that existence of an $\omega_1$-sequence of reals is equivalent to the weaker assertion that that there is a choice of injections $(\varphi_{\alpha})_{\alpha<\omega_1}: \alpha \rightarrow \mathbb{R}.$