Feel for compression. if I'm going too slow when I drop my clutch, my back tire skids from the resistance from compression in the engine. Can you feel the resistance? back tire skid, or just slowly coming to a stop, like dragging your feet. Probably the easiest compression test.When you do try turning your pedals and see if the piston goes up and down easily. If the answer is no, then that'll be a good indication that there's been damage in there.
I can't speak for running out of oil in these specifically, but Ive seen quite a few 2 strokes survive without oil for a bit. No guarantee its shot yet. Ive had some really boned compression readings on engines that will still run. This is a fine engine to learn on, parts are cheap, operation is simple. You'll be a pro in no time and back on the road.
Best I can tell from your pic, is that someone used better hardware on your head, and maybe not better connectors for your magneto.
maybe I missed it, but regarding your choke... take off your air filter, have a look. You will see your choke. When "on" the intake will be blocked or partially blocked. FWIW, I have not needed to choke unless its <40 F As I remember up is off down is on.
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