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 the point in a small ball and computes the surface area of the stereoraphic projection of the figure on the corresponding sphere. This provides a scalar function whose level surfaces give 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.
Added as an edit. The corresponding two dimensional problem was very popular in classical differential geometry. An isoptic curve (other names---equioptic, isogonal) to a figure (not necessarily one dimensional) is the set of points from which the figure subtends the same angle (not necessarily a right angle). An interesting question is when these curves are equipotentials, i.e., the level curves of a harmonic function. This is not the same as asking whether the function described above is harmonic. This question is motivated by the fact that this is indeed the case for some of those few cases where the isoptics have been computed (e.g., the isoptics of an ellipse).