ad Q1. It is a simple, if tedious, elementary exercise to compute the solid angle subtended at a point in space by a figure in the $x$, $y$ plane as a function of its coordinates. One places it in a small ball and computes the surface area of the stereoraphic projection of the figure on the corresponding sphere. This provides a potential function whose level surfaces provide an answer to Q1 in quite a general situation. Of course, the computation of this function involves double integrals over the given figure and it will depend on the latter whether this can be done explicitly.
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