If the fridge is on when the car is off either you have a faulty relay or the wiring is wrong. It has nothing to do with the strength of the battery and I would bet that if you left it for a couple of hours and tested the voltage of the car battery it would have diminished somewhat. A caravan fridge does NOT work from a caravan battery, it's either 240v or gas when static and 12v only when being towed. Test the pins as someone suggested and you will have the answer. Regards
Oh the lights, on my van if the caravan is connected and the car is running only the battery charger and fridge are working, they are on the same circuit. Engine off and caravan still connected everything 12v works in the van, lights and flushing toilet (being powered by the caravan battery) however on my older van the plug had to be disconnected from the car for the 12v lights to work, hope this helps