Theorem 3.2 in Quantitative stability of variational systems: III.ε-approximate solutions Hedy Attouch, Roger J. -B. Wets kind of gives an answer for epsilon-optimal solutions.

@RobertIsrael @Dirk; Thanks for the answer and the comments. It was helpful. I think this shows that there isn't a nice Lipschitz-like bound for any two functions, but uniform convergence of convex functions does still imply convergence of the derivative -- isn't this what Theorem 25.7 in Rockafellar states?

I was hoping Hausdorff distance would be easier to handle; and I wanted something where there wasn't a dependence on the $r$. I think in that paper if I choose $r$ to be large enough, since $\partial f(x)$ and $\partial g(x)$ are bounded, the metric $d_{r}$ becomes the Hausdorff distance, but this isn't uniform in $x$, which is something I need.

Thanks Dirk. I tried looking, but would you know whether there has been some work in this area recently? I can't seem to get anything quantitative beyond the 1993 paper I linked above.

Other than the Jean Paul Penot paper, nothing else seems to come up with a quantitative estimate using the Hausdorff distance.

About the answer to Q1, I don't know if the proof is as direct as that, because that kind of proof works only if $x_{V}^{*}$ and $x_{U}^{*}$ are close but not the same distance to $y$, but otherwise, if they are exactly the same distance to $y$, the proof might be more subtle.

Thanks Christian! What about the case of taking the supremum over all $y$? So $\sup_{y\in \mathbb{R}^{n}}\|x_{V}^{*}(y)-x_{U}^{*}(y)\|$? Do you think the same convergence holds?