Electrical – Melted neutral wires in a junction box

The ground conductors were probably functioning as your neutral return (just not very well) until you disconnected it at the light. So not only do you have an open (or high resistance) neutral wire, but you also have too much resistance in a ground wire. If you've checked all the boxes and terminals for loose connections, you may have a damaged wire in the structure somewhere. The high resistance will cause heat (and risk of fire) if you continue to use the circuit. I would probably contact someone who has access to test equipment that can diagnose the actual in-wall wiring. (like maybe an infrared thermometer to detect a hot spot, or a voltage detector to trace wires in the walls, or at least an ohmmeter)

The AC input circuits of some devices use a power line filter circuit that works to keep high frequency stuff inside the equipment and prevent fast spikes from entering the product from the power lines.

The schematic of a typical power line filter looks like this.

The two capacitors that you see in the filter from line side to earth and from the neutral side to earth can form a voltage divider for AC voltage and place approximately half the line voltage on the earth wire. This would be the case for instances where the earth ground connection was open circuit.

Normally this is not a shock hazard issue because the neutral wire is shorted to the earth ground connection at the power panel.

The fact that you found this AC signal present demonstrates that some part of your house wiring has a open earth ground wire someplace. For the safety of your self and your family it is essential to get this fault corrected. The floating earth wire is an invite to it not performing its safety function when you least expect to need that safety feature.