@n901 All reasonable formulations of "all sets of reals are Lebesgue measurable" are equivalent in ZF, and prove the $\sigma$-additivity of $\lambda$ (see mathoverflow.net/a/393162/109573). We get a relative consistency proof over ZF by building the Solovay model relative to the Levy collapse of $\omega_{\omega}$ and adapting Solovay's argument to show $\lambda^*$ and $\lambda_*$ agree on subsets of the unit interval.

I expanded the first part. Does the second part address what you were looking for regarding whether the players "almost always fail"?

I have a strong interest in this question. Proof strategies for several problems I've worked on are hung up on verifying this conjecture or minor variants of it.

Yes, that fails for the indicator function of an infinite Dedekind finite set $S:$ globally sequentially usco, sequentially continuous off $S,$ discontinuous at the condensation points of $S$ (which is a perfect set so is not contained in $S$). If you demanded $f$ be in some Baire class AND usco, it would be significantly harder to find a counterexample.

Has anyone checked if every set can be the image of a dually Dedekind-finite set?

Regarding the cases mentioned in the last paragraph: $(2^{\omega})^L$ (as well as any $\omega$-model of $\mathrm{RCA}_0$) is either measure 0, maximally nonmeasurable (inner measure 0 and outer measure 1), or equal to $2^{\omega},$ since it is closed under dyadic translation and from a real $r \not \in A=(2^{\omega})^L,$ we can construct infinitely many disjoint translations of $A.$

@GabeGoldberg Assume $V=L,$ let $\kappa$ be inaccessible, let $G=\langle G_{\alpha} : \alpha < \kappa \rangle$ be Levy collapse generic. Let $R=\mathbb{R}^{L[G]}$ and define $f: R \rightarrow \kappa$ by setting $f(x)$ to be the least $\alpha$ such that $x \in L[G \restriction \alpha].$ Then $M=L(R, f)$ has $(R, \prec)$ as in the question, and every set of ordinals in $M$ is OD from $f$ and some $G \restriction \alpha,$ but $R \cap HOD^M(f, G \restriction \alpha) = R \cap L[G \restriction \alpha]$ is countable in $M.$ So no subset of $\omega_1$ codes an injection from $\omega_1$ to $R.$

Actually, from the equivalence of 4D smooth and PL manifolds, can isomorphism between two 4D smooth manifolds be expressed combinatorially via triangulations $T_i$ of $M_i$? Something like “there are simplices $s \in T_1, t \in T_2$ such that for all $n,$ there is a combinatorial isomorphism from the set of simplices within $n$ adjacencies of $s$ to those of $t$”? I’m out of my depth here geometrically, but that could confirm the equivalence relation to be Borel.

Naively this equivalence relation is analytic, because it posits the existence of a diffeomorphism. I don’t think there is a simple trick to proving this conjecture. In the 4D case, it will probably requiring actually overcoming the longstanding problem of finding nontrivial invariants for smooth structures.