Am I Getting the Most Out of a Refrigerator?

Just go over the thing with a magnet, and whatever doesn't attract the magnet, pull. That will be all the copper tubing and hopefully the condenser (sometimes these are prefab steel). Fan blades and interior pieces will sometimes be aluminum. Also pull the motor and take it apart if you find it worth your time, or sell for motor price.
Magnet is your best friend.

The ice machine has cast aluminum. Also sometimes if the fridge has metal catch guards on the doors (where you'd put condiments etc), those are aluminum too.

On some fridges, on the upper tops of the fridge and freezer doors I have pulled off aluminum. It is thin stuff and is screwed on to the light iron.