I haven't followed the whole story here, just happened to see you mention the idle solenoid. If it's anything like mine, sounds like it's working as it should, shutting off the idle fuel feed as soon as power is disconnected. It is to prevent dieseling after you turn the engine off.
The solenoid should get power whenever ignition is on, to enable idle fuel feed. So you adjust idle speed with the solenoid energized. It's supposed to kill the engine when power goes away. It's not a high/low idle speed solenoid.
