5. In order to have a chance to regulate the temperature well - and not keep cycling through blasts of heat and cooling - they'll need multiple temp probes, and an awareness of the outside temp and humidity as well, since ceramic insulation or no, the external environment will play a huge factor.
If your PID controller is cycling like that something is seriously wrong. Even a poorly tuned controller should eventually stabilize, unless the gains are way out from where they should be. The controller should be easily capable of overcoming fluctuations in ambient temperature, even in New England.