Yes. Yes. Yes. Yes.
I'm assuming you mean the plug in the back? That controls when the alternator is charging or not? No, I haven't checked that. Like I mentioned, I am planning to install the diode to increase the charging voltage, so I will inspect those wires this week when I do that.Have you checked the connector at the alt? IIRC one of the wires going into the alt to the regulator causes the light to pop on based on grounding etc. Might be a short or break in that wire.