Should I take my iPod out of a case before (re)charging it?

Yes. Apple recommends that you take the iPod out of any case before (re)charging it.

The hard drive won’t spin when charging from the wall or the car, but it still gets hot inside.

1. The iPod has no internal cooling fan and no vents, so relies on its stainless steel back to act as both a heat exchanger and a heat sink. 
2. Any heat exchanger relies on having maximum continuous surface area in contact with the cooling medium (in this case ambient air).

   The continuous part is important, because it means that cases with ventilation holes still dramatically reduce the heat exchanger’s efficiency. Having the steel back in contact with an insulator (a case or jacket) is a bad idea.

3. Then iPod is full of narrow guage, minimum insulation wires and flimsy soldered connections which will become brittle or may fail, if regularly subjected to high temperatures.