If I understand correctly what a "harmonic function on the cone" is, there is no difference
between the cone and the disc for the Dirichlet problem.

This cone is a surface equipped with a Riemannian metric (induced from $R^3$) with one singularity (at the vertex). The metric defines a conformal structure turning the cone minus vertex into a Riemann surface. It is easy to see that this Riemann surface is conformally equivalent
to the punctured disc. The conformal map is easy to write explicitly. Now it is easy to solve the Dirichlet problem on the punctured disc (with data on the circle):
the solution is the same as for the usual disc; it extends to the puncture by contnuity.
Thus there is no difference between this cone and the disc for the Dirichlet problem.

Concerning the Green formula, it also holds, but notation in the question seems strange:
what is $dxdy$ and $dS$ ? One has to take the gradiant with respect to the metric on
the cone, and write the area element on the cone instead of $dxdy$...