I don't know if you can have something as fancy as Thévenin's theorem in a general Riemannian manifold, but as a physicist I'd say you'd be better off looking at generalizations of the conductivity, $\sigma=1/\rho$, where $\rho$ is the resistivity and can be measured as $\rho=R A/L$ for a (homogeneous) wire of cross-section $A$, length $L$, and resistance $R$. Then you can formulate a local version of Ohm's law as $$\mathbf{j}=\sigma\mathbf{E},$$ where $\mathbf{j}$ is the current density, related to the current $I$ flowing through a surface $\Sigma$ as $I=\int_\Sigma \mathbf{j}\cdot\textrm{d}\mathbf{A}$, and $\mathbf{E}$ is the electric field, related to the potential difference $V(\mathbf{x})-V(\mathbf{y})$ between arbitrary points $\mathbf{x},\mathbf{y} $ as the line integral $\Delta V=\int_{\mathbf{x}}^{\mathbf{y}} \mathbf{E}\cdot\textrm{d}\mathbf{r}$ (independent of the integration path).

As for the second question, you'd have to solve Laplace's equation $\nabla^2 V=0$ on a rectangular-box domain $[0,L]\times[0,L]\times[0,h]$, with appropriate boundary conditions - I'm not that sure there, but I'd set $V=0$ on $(0,0,z)$ and $V=V_0$ on $(L,L,z)$ to begin with; if that does not determine a unique solution then Neumann conditions on the rest of the boundary should do it. You then find the electric field $\mathbf{E}=-\nabla V$ to get the current density and integrate across an appropriate spanning surface, say the other diagonal plane.

I'll try and flesh this out and fill this in when I have time.

EDIT
J. Cserti's paper Application of the lattice Green's function for calculating the resistance of an infinite networks of resistors, *Am. J. Phys.* **68** no. 10, pp 896 (2000), doi:10.1119/1.1285881, arXiv:cond-mat/9909120, solves the discretized problem of an infinite network of resistors on a square grid (including as a special case the Nerd Sniping xkcd problem). In the continuum limit that yields a solution to this problem.