A spherical polygon is "a closed geometric figure on the surface of a sphere which is formed by the arcs of great circles." (from Spherical Polygon). What I would do -- since there are an infinite number of rectangles of any given side centered on a given point -- is find the intersections of the great circles 1km north, south, east, and west of your point.

For question 2, try Spheres: Miscellaneous Algorithms. They do mention that it can be rather difficult. If your rectangle is oriented so its sides are exactly east-west and north-south, it's much easier.