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.