For your first question, I don't know for sure but we called these "counting problems" when I worked with lattice models in protein structure prediction. See https://en.wikipedia.org/wiki/Lattice_protein for context and we spent a lot of time eliminating solutions that were identical after rotation, and self-intersections (there's a suprising amount of exciting math associated with determining if a chain specified using local angles intersects with itself in absolute 3-space.