At the last school I worked at it was common practice to allow a student to start the engine on their own. The instructor would then walk out and climb in with it running. This would typically be a couple of lessons before first solo, and signed off as capable of solo engine start in their student records.
Solo taxi would be fine, if you're at an airfield where the aircraft are parked a long way from the pumps (not dragging distance) or to taxi an aircraft back from the maintenance hangar. No exams or laws to worry about, but if the student hits something while taxiing solo, it's going to be the instructor getting in trouble just as much as the student.