The answer to the first question is no, at least if one is willing to assume large cardinals in $V$. I'd have to research what the optimal hypothesis is. It is shown in the two Neeman-Zapletal papers that if there are infinitely many Woodin cardinals below a measurable cardinal and above some cardinal $\kappa$, then forcing with a c.c.c. partial order of cardinality less than $\kappa$ cannot change the theory of $L(\mathbb{R})$ with ordinal parameters. Since the cardinal invariant $\mathfrak{t}$ (the tower number) can be made arbitrarily large by c.c.c. forcing, this shows from the same hypothesis that the intersection of any wellordered collection of dense open subsets of $\mathcal{P}(\omega)/\mathrm{Fin}$ in $L(\mathbb{R})$ is dense open. For this specific problem, we need to intersect only $\omega_{1}$ many dense sets. Since c.c.c. forcings don't change $\omega_{1}$, we can do with the ostensibly weaker hypothesis that c.c.c. forcing doesn't change the theory of $L(\mathbb{R})$ (not allowing for ordinal parameters), which gives that intersections of $\omega_{1}$ many dense open subsets in $\mathcal{P}(\omega)/\mathrm{Fin}$ are dense open. I don't know if this can be proved just assuming $\mathrm{AD}$ + $V = L(\mathbb{R})$, or even whether one can deduce it from the Henle-Mathias-Woodin result.

Suppose now that $\langle \tau_{\alpha} : \alpha < \omega_{1} \rangle$ is a sequence of $\mathcal{P}(\omega)/\mathrm{Fin}$ names for nonempty sets of reals. By the assumptions of the previous paragraph, there is an infinite $a \subseteq \omega$ such that for each $\alpha < \omega_{1}$, the set $B_{\alpha}$, consisting of those reals forced by $a$ to be in $\tau_{\alpha}$, is nonempty. The sequence $\langle B_{\alpha} : \alpha < \omega_{1} \rangle$ is in $L(\mathbb{R})$. If $a$ forces each $\tau_{\alpha}$ to be a mod-$U$ equivalence class in the Baire space, then the members of each $B_{\alpha}$ must agree mod-finite on $a$. So, restricting to $a$, we get an $\omega_{1}$-sequence of disjount countable subsets of the Baire space, which contradicts the conjunction of the Perfect Set Property and the Baire Property. So there are no $\omega_{1}$-sequences of distinct mod-$U$ equivalence classes in $L(\mathbb{R})[U]$, regardless of the relationship between the classes.

A similar argument gives the Perfect Set Property in $L(\mathbb{R})[U]$, which was originally proved by Di Prisco and Todorcevic. It seems that the same approach should give a negative answer to the second question, but I don't see it.