These old CJs can be difficult to work on when wire colours are changed.
As you can see in my ignition diagram, the light green wire on the coil goes to the Ignition Module. The red with white tracer wire at the coil is from the ignition circuit.
If you disconnect the wires from the coil (they need to be disconnected to measure, otherwise you are testing both together) the red with tracer should have about 7V (dim bulb) in Run. In Start, if the solenoid worked, it would become a Bright bulb with 12V. This is not effected by the NSS.
The light green wire at the coil is the switch from the module. What it does is ground the coil in time with the engine. It has no voltage, it is a ground. If you have it disconnected and the test bulb is lighting up there is something wrong.
To test that light green wire is giving the signal you would hook one end of the test bulb to the battery positive and the other to the wire. As it is switching from ground to open circuit you will see the bulb flash when/if your engine turns over.
However none of the above will effect the operation of the solenoid.
The Neutral Safety Switch in your auto box is just that, it is only closed when the box is in Neutral and Park. It is cunningly wired in the light blue wire to your solenoid. It does not appear in the diagram I posted earlier. However, if you have battery voltage, in Start only, at the S tab (light green or blue) in Park and Neutral it is working fine.
It sounds like a grounding problem. When your Dad (Dads are always right) was pounding the battery terminals was this because they were old, corroded or just a bad fit? It sounds like the solenoid is not grounded back to the battery. This may also explain why other systems are flickering.
There should be a large black battery cable going to the engine, often fixed to the engine block but sometimes the starter holding down bolt. There should be another strap from engine to frame. CJs are notorious for bad grounds, some twit considered that grounding a major component to a rusty steel fender which could connect to the tub which could connect o the frame which could........ I think you get the picture, electrons flow around a circuit of which only one half is in nice plastic coated copper wires and the other half is through a muddy rusty bit of steel.
The inside of the battery cable can corrode and fray, they are not indestructible.
I have a seperate ground system that links major components via copper cable back to the battery negative. You may want to try grounding the solenoid with a wire back to the battery negative and see if it helps.
As to the reverse light I believe it is a different switch on the transmission. the reverse lights I believe are powered and the switch just grounds it. If so you can get under it and find the switch you should find voltage there.