2: Any three points in 3D space lie on a single plane. If a third vertex lay on this plane then the fourth point must also be in the same plane for its centre of mass to be on that plane (otherwise there would be a component of mass in a direction perpendicular to the plane), making it completely flat. A flat tetrahedron would have no centre of mass as it would have no mass, so this cannot be the case. As a result only the two vertices we chose can line on the plane.
5: Yes: start half a km away the North Pole. This way the movements east and west both occur on the same line of latitude and so cancel each other out. Of course, this assumes that you keep the same direction when crossing the Pole even though your north and south would flip.
6: Consume all four pills