How can I diagnose disturbing voltage potentials?
From all this, I conclude the following is likely:


Only the outlet in the kitchen next to the sink is properly wired.
All other outlets have hot and neutral reversed.
There is no ground connection.
Unexpected observed voltages (mostly 60 volts) are caused by stray capacitance.


I'll mark it accepted when I get complete confirmation.

Outlets need to have their hot and neutral corrected. Outlets need to be grounded or labelled as "GFCI protected" and "no equipment ground."

Fluctuating voltage from the actually hot line to true ground (the water pipe) is caused by current flow reducing the electrical resistance of the line. Voltage is otherwise high and stable for the capacitive currents as they don't induce enough flow.