The more you load up a lighter socket, or the single wiring that powers it, the more likely it is to fail on you. Most cars have the bare minimum wiring for the lighter, they are only good for 15 - 20 amps, look on the fuse box to see what yours is, but regardless, they don't like a big load on them. Even then the light wiring doesn't like big loads on them, running a multi point adaptor to run phone chargers, gps and the like is one thing, they only draw a small amount of power, but a fridge sucks heaps of power.
As suggested, look at a Waeco or Engell, I have a Waeko with a RAPS kit which hard wires to your battery and has a dual outlet, one for the fridge and another for anything else. It can be set to only supply power when the car is running, or set to keep power to it until the battery can no longer support it and only has enough power to start the car again.
As for adding extra lighter sockets in the back, again dont donkey them off the existing socket, run new heavy wiring for each of them direct from the battery, each with its own fuse. This will save you from having failures (or fires) in the long run.
If you want a 240V power point, get an inverter kit from a 4x4 shop to set it up. you can choose the power output according to what you want to run from it. Fridges suck a lot of power so it would be cheaper and more convenient to get a portable fridge like a Waeco. Ours is 80L, fridge/freezer, with RAPS and backup battery that runs the fridge for 30 hours stand alone and recharges through the RAPS kit or 240V, came as a kit with thermo cover, mounting kit etc etc for about $1500