May not be a battery problem.
Check the voltage on the battery with engine stopped then check the voltage with the engine running. Should be about 14 volts with engine running. If it isn't, get the alternator checked.
You are getting too technical.
Try what BM Chidders suggested and work from there. You can also test it with a resistance meter.
If you are going to remove the gauge then either have a new one or a second hand one with you to change it over.
Presumably the heater is working so that would rule out the thermostat?
Quite possibly has come off the runner. Remove the door card and check it out if you feel competent enough. If not, take it to a garage or an auto electrician.