Taking the caravan alone, if the mains hook up is disconnected, then the Control part only of the fridge will be powered by the caravan battery. The fridge will not cool off the caravan 12 volt DC supply alone, (or any other 12volt supply for that matter when not connected to a correctly wired vehicle) it requires either the gas supply connected or mains electric for it to cool. The caravan 12 volts system always powers the control side of the fridge (either from the battery or PSU if switched on when mains hook up is connected).
Similarly, when connected to a correctly wired vehicle (without the engine running) the fridge control and some interior lights will be powered by the permanent 12v supply (pin 9 of 13 pin socket). If you have a Touareg without the extra auxiliary wiring fitted (not an uncommon event, especially with a brand new Touareg fresh from the factory with a factory fit towbar). Coincidentally, my brand new Treg is having the extra VW auxiliary wiring fitted today by the supplying dealer.
If you have a correctly wired 13 pin socket on your car, when you attach it to the caravan, without the engine running (doesn't matter if the ignition is on or off) the caravan controls will light up together some of the interior lights if they are switched on. Start the engine and when the car deems it safe (it prioritises the vehicle battery charging system) then 12volts will be supplied to power the fridge so that it will cool using the 12 volts vehicle supply on pins 10 and 11.
Any other type of wiring is dangerous and can damage the car's electrics and prevent it charging the battery (fitting a relay to allow the fridge supply when ignition on etc), it must be done via split charge relay sytem fitted by someone who knows what they are doing as ESP, ABS, Alarm and other safety systems are affected by connecting a trailer ESPECIALLY WITH ALL MODELS OF TOUAREG. A properly wired Touareg will sense the connection of a traier and make alterations to ABS and ESP to employ anti-sway, modify the alarm system so that it sounds if trailer disconnected and switch off the car rear fog light, there's also some bulb failure stuff going on.
Exactly the same circumstances exist if you have twin 7 pin sockets fitted.
So, connect the car, no ignition on or engine running, you should have fridge control lights on and some interior lights available. Start the engine and after a short while (you may have to raise rpm to say 1500 rpm), then the fridge should start cooling and the caravan battery should be charged.
If you have a reasonably modern Dometic fridge (most caravans seem to) then, when the fridge has only permanent 12 volts supplied for the control circuit (either from car or caravan battery) then you will see a flashing blue LED on the control panel. As soon as any other cooling supply is available (gas, mains electric or vehicle 12 volts) and cooling the fridge (you will have to select this by the appropriate button) then the LED will stop flashing.
HTH.