Is the range for the pot always 0 to 4095? I know it differs with various battery voltages, so does this mean, if I have lower battery, it will be 0 to something less than 4095?

Potentiometer values should stay the same no matter what the battery voltage is.
The values realistically go from about 5 to 4090 because of mechanical stops, but I haven’t read anywhere that differs, so I’m not sure where that’s coming from.

Hmm. That would be a good experiment to run. I want to say it should not matter as the potentiometer is measuring voltage drop through variable resistance.

So if the source voltage drops does it make the potentiometer read differently? Good question!