Yes you can start the engine via the pull cord with the key in the ON/RUN position.
The "Fuel Primer Solenoid"... Manual mode = turn the RED lever so it faces away from the solenoid. Now, when you press the fuel primer bulb (and when the engine is running), fuel will flow thru the solenoid into the intake manifold to prime the engine. BE SURE to turn that RED lever back so that it faces the other end of the solenoid after the engine starts as that facing away position will flood the engine.
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The neutral safety switch is usually a little flakey so to speak... and simply loosening the screws ever so slightly, just enough to allow you to slide it closer to the cam that presses against it... and doing so usually corrects the problem.
Testing it is just a matter of making sure you have 12v at the "B" (battery) terminal of the ignition switch... and that same 12v at the "S" (solenoid) terminal of the ignition switch when in the START position... which should allow that 12v to be now flowing thru the neutral safety switch (check it).
By the same token, if everything else is okay, the engine should be cranking over when in the START position.
Let us know what you find.