Theorem 5.10, pp.57 from Villani's Old and New is the relevant theorem for existence of maximizing potentials. But "measurability" of the potentials comes more readily as consequence of (Twist). In OT i think it's more useful to be specific about your costs and your measures, and not attempt to maintain "max generality". For example, measurability of what potentials on what type of spaces relative to what type of costs? The regularity of the potentials is typically "inherited" from the regularity of the cost function $c: X\times Y \to \bf{R}$.

I don't think anybody knows how to prove "from foundations" the measurability of the c-concave potentials. But it's better to specialize to costs which satisfy (Twist) conditions, then you more readily obtain the uniform semiconcavity, local Lipschitz constants, etc, of the potentials. In practice it's better to first obtain the almost-everywhere differentiability of the potentials by differentiating along the graph of the c-subdifferentials, from $c(x,y)=-\phi(x)+\psi(y)$ to $\nabla_x c(x,y) = - \nabla_x \phi(x).$

Aren't you redefining the barycentre of the measure $\mu$? In that case an implicit formula for the barycentre ($x_m$ in your notation) is $\int exp_{x_m}^{-1}(x) d\mu(x)=0$ in the tangent space $T_{x_m}{\bf{R}}^n$. And that's basically the equation you found. A reference is J. Jöst's "Nonpositive curvature: geometric and analytic aspects" (Chapter 3). The point is you can find $x_m$ coordinate wise, i.e. $(x_m)_i=\int x_i d\mu(x)$. That's a direct formula. The average coordinate is the coordinate that minimizes average distance.

The problem with this question is that $\chi(M)=0$ for many (most?) manifolds $M$, and the Euler characteristic $\chi$ is not a strong topological invariant. So the right hand side function $h$ will not be uniquely defined, especially on $h(0,0)$.

Lemma 7 of your paper is interesting. Do you find Qi's proof satisfactory?

@leomonsaingeon If the potentials are locally Lipschitz on $\Omega$, and their restrictions to $\partial \Omega$ are also locally Lipschitz, then Alexandrov theorem (applied to $\Omega$, $\partial \Omega$) would give a.e. uniqueneness with respect to $\mathcal{L}^d|_\Omega$ and $\mathcal{L}^{d-1}|_{\partial \Omega}$. I'm sure that's obvious to you, and therefore the gradients of the max Kantorovich potentials are a.e. uniquely defined by the OT. But pointwise uniqueness of $\phi, \psi$ is more difficult, and I don't know any references.

You have uniqueness of the Monge OT plan $\pi$, and in your setting the uniqueness of the subdifferentials of the dual Kantorovich potentials $\phi, \psi$. So why trouble yourself with uniqueness of the potentials themselves. Honestly why? It's very complicated question and IMO a waste of time. Are you just curious, or looking for research project, or do you have an actual application in mind? I would politely ask Jun Kitagawa or Robert McCann or Brendan Pass, if they know the answer. I've never seen them on MO.

I completed my PhD and wanted to enter industry, except i'm still looking for work and spending more time than ever on mathematics. I'm working to publish, and not ready to perish yet.

I think you need to assume $M$ has finite volume to be more specific about the cusps, which are noncompact and not contained in the convex core (which is compact). The cusps are basically complementary to the convex core. Have you reviewed Thurston's Three Dimensional Topology book? Or review the example of modular surface $X=\bf{H}^2 / PGL(\bf{Z}^2)$ in detail, which motivates the whole idea of the structure of cusps.

i don't know, but i would ask Prof Young-Heon Kim at UBC.