On further inspection, a bunch of the comments here seem to have been deleted, so that example is now rather contextless.... but the point of it really is that the OP's conjecture is false.

@gaoqiang The Harnack inequality definitely has applications for minimal hypersurfaces, provided that you have an a priori bound on their extrinsic curvatures, so I would not call it "almost useless". But as I already said, the OP's original conjecture that its constant only depends on n is false -- that was the whole point of including that example.

For each $\delta > 0$, do you have any estimates on the "size" on the set $\{(x, y): |f(y) - f(x)| \leq (1 - \delta) |x - y|\}$ assuming that $f$ is eikonal? It seems to me that such a set must be "small", in particular it feels like it should be first-category.

@V.Moretti OK, I think what I wrote now is correct (and hopefully clearer, but let me know if you have any questions).

@V.Moretti I think I was mistaken, and your condition only shows that the lower (modified, maybe?) Minkowski dimension is at most n - 1. Now it's not true that the lower Minkowski dimension has to equal the Hausdorff dimension, so I think what you conjectured is false anyways, but I'll have to work through the details more carefully tomorrow if someone else doesn't answer it first. Sorry for the confusion. :(

The Thurston metric generates the same topology as the usual topology on Teichmueller space, so while I might be misunderstanding the question, I think it is equivalent to: given $x_0$, under what minimal/generic conditions must a compact family of self-homeomorphisms of $\Sigma$ not map $x_0$ to itself?

Sorry, I forgot to respond -- in the application, $A$ is countable, and can have accumulation points. (It turns out that much later in the paper, Bonahon proves that his sum absolutely converges under much stronger hypotheses, which are the ones that he actually uses. So maybe the answer is that "the sum in question is undefined without those hypotheses".)

I don't know about a Feynman-Kac formula specifically, but the $p$-Laplacian has a nice probabilistic interpretation which generalizes the Laplacian's interpretation as the infinitesimal generator of Brownian motion, see arxiv.org/abs/math/0607761