[oh'mike]

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.

 

[COLDIRON]

I would definitely install 1/4" copper rather than any flex tubing and do what oh'mike said.

 

[oh'mike]

Nice plastic boxes with shut off valves are available---allowing you to recess the shut off behind the fridge---then add a separate line from the box to the fridge.----looks nice and is handy.

 

[Leah Frances]

Use the search function on this site. I did this project.

 

[Improvised]

I would have never guessed this myself, but I guess RO systems are corrosive to copper piping. I guess I'll have to keep it plastic or stainless steel.

Source: http://www.allgoodwaterfilters.com/10/2006/copper-pipes-for-reverse-osmosis.html