The solenoid's actual name is anti-afterfire solenoid. Its purpose is to stop the flow of fuel at the same time that ignition is turned off so that the inertia of the engine doesn't pull fuel/air through the engine as rotation stops and result in an afterfire. You probably won't experience an afterfire anyway but you an mitigate by letting the engine idle and cool a minute or so. Besides, afterfires don't hurt anything.
See
THIS EXPLAINATION. (the solenoid purpose fits in the delayed ignition category)
And yeah, I know the pin moves freely. Until it doesn't and plugs you up. When that happens flip the choke on and take your time until you get to the tools and replace the solenoid with a bolt. (or do the snip). (bolt's cleaner
)
The spark plug doesn't look all that bad. Maybe a bit lean. Might need the 106 for 8K. (I
knew Argo was being conservative with their jetting but didn't have the time to do the experimentation.)
But my reading: The base ring has soot that I would guess means that idle is too rich. The insulator is off white and the electrode doesn't look too rounded so at higher than low idle (the high idle circuit of the carburetor - it has two circuits) it is running somewhere pretty close to where we we want. Do google "reading spark plugs" though as this is not my science.
Your pushrods look good. Like PaSean said, when they are bent they fall out from under the rocker.
You probably need to adjust low idle mixture at higher elevation and you probably need to look at your clutch tuning.