Most fridge lines are 1/4 inch---If you run it in the walls consider 1/4 copper and add nailing plates so you don't hit it with a screw while installing the cabinets.
Another method is to notch the bottom of the cabinets and run the line under the boxes.
I can't help with the insulation---never done that one.
Another method is to notch the bottom of the cabinets and run the line under the boxes.
I can't help with the insulation---never done that one.