Reverse Physics and a Car
1. The wording here is slightly misleading. Newton's Law actually only applies to point masses, and this is what is meant by "all bodies" in the first paragraph. However, to get an example, we obviously can't use a point mass.
There are many things that would work. The general idea is to have the bulk of the mass away from the center of mass, in a symmetrical manner. For example, a ring of some sort would work.
At a large enough distance from the ring, Newton's law approximates the gravitational attraction just fine. However, as the other object gets closer to the ring (the second object can be a point mass, another ring wouldn't necessarily work), the mass from one side of the ring cancels out the gravitational force from the other side of the ring. As the point mass approaches the center of mass of the ring, the overall gravitational attraction approaches 0.
2. The problem with the claims of the mathematicians is that in order to get a distance (the solution), you can't multiply two velocities, or two times. The graph is only a semicircle because we have arbitrarily made the scales used for time and velocity such that it looks that way.
In order to get the correct distance, we need to treat the graph as an ellipse (or rather, half an ellipse). Then, the area is (pi/2)*(t/2)*v = (pi*t*v)/4, which is the distance travelled (and it has the right units!).