I think that the one breaker seems to control the problem is a major hint. The problem almost has to be on that circuit.
Enough current to melt the no-hub connectors? I'm impressed. I suspect that neighbor has a few problems. Here are some ideas:
First, I'd find out what's on the circuit. Then, I'd see if the problem persisted with every appliance unplugged. This removes the appliances as the source of the problem. (I don't think they are, but let's stop guessing, and be systematic).
The next step is finding the place in the circuit where the hot contacts a pipe, the conduit, or whatever. A cheap megger is a good tool to use to check the wires.
You notice I am not thinking "neutral." Not, at least, if the problem persists without any appliances plugged in. Since you claimed to measure 120v at one point, I don't think that current is flowing through any load.
Once you find the short to ground, you've found half the problem. The other half? Grounding.
I suspect that the neighbor has a good water bond ... it might even be to the sewer line. I also suspect that your customer has a poor one. I doubt either house has a ground rod.
Keep this in mind: The only way the piping can become energized is if there is no path for the electricity to get back to the PoCo transformer.
