Hi Nick2,
We are also fairly new to caravanning, and on our first 'van, but I will try to help you based on our 'van.
No, you don't need any of the caravan electrics on to run the fridge from the car. When the engine's running the leisure battery will charge and the fridge will work (assuming the fridge is switched on).
Secondly, assuming your heater is similar to ours, then it is perfectly ok to use on hook-up. Our heater (can't remember the make off hand) will provide blown air on hook-up or can be used as a gas convector heater with or without a 12 volt fan.
Hope this helps.