Don't assume anything about wires based on size.  You need to see where they go and whether they're connected to the 12vdc system or the 110vac.  Misconnecting them could lead to some rather inconvenient consequences.  You should see a 110vac wire/plug coming out of the refrigerator and an ac plug in the area behind the refrigerator where you can plug it in.  That operates the ac side of the refrigerator.  The 16 +/- gauge wire is for the dc for lights and to facilitate gas operation.  The 10 gauge wire is for running the refrigerator off your batteries, which you should only do for very short times or you rapidly drain your batteries if they are not being adequately charged simultaneously.  

I replaced the Dometic refrigerator in our 172 and it was an easy project.  I bought the refrigerator from the rPod factory and they were very helpful in helping with the needed wiring modifications to correct a factory deficiency in the original installation.  

One other thing you need to be very careful about is making sure the back of the refrigerator area is well sealed.  There is a gasket there that takes care of most of it, but you should caulk the places where the gasket doesn't seal well.  Also, while you have the refrigerator out, you can add some insulation on the walls and above the refrigerator that will help keep it a little cooler.

Good luck with the project.