Is it possible to use the water line for the washer to add a line for a fridge.

Sure. I have seen flexible copper or plastic, how about from sink?

Sure it possible. You can tap into any cold supply line. Many refrgerators use a saddle valve onto an existing line. Connect a saddle valve to copper pipe - YouTube (http://www.youtube.com/watch?v=FB76LXrYKsU)