get rid of the 12 volt thermometer!
here in the us volts vary a bit in homes. if you put your meter into wall outlets, the volts will vary a bit.
12 volt dc is direct current, this may be causing some of ther problems.
try a battery operated thermometer.
try a new heat pad.
on 120 volt circuits, it doesn't matter which prong is the big one. here in the us, almost everything can run on either polarity. in the old days it mattered.
vaughn