Chris Y,
Yes, that is correct on a 4-stroke engine (Intake (Dwn) , Compression (Up), Power (Dwn), Exhaust (Up). (Up & Down = direction of travel for the piston). The points are opened and closed by the camshaft. The crankshaft makes two complete revolutions while the camshaft rotates only one revolution. The points should open about 20-degrees before Top Dead Center (TDC) which is the timing mark "S" centered in the viewing hole. When the points open the spark plug should fire.
When you take the spark plug out of the head, ground it to the frame and crank the starter, do you see a bright blue, regularly repeating spark? If you do then the ignition may be fine.
Next I would look at compression. Have you checked the gap on the valves to make sure they are fully closing? You can check this by removing the Carb and then the valve cover assembly to gain access. Make sure the inner pan of the valve cover assembly has the little drain hole at the bottom. Get a feeler gage and check the gaps to make sure they are per the specs in the manual. Also rotate the engine by hand and observe the valves to make sure they go up and down smoothly and don't hang up. If the valves are closing properly, then you are not getting adequate compression to fire the engine other than the occasional backfire.