Can I run the 2″ drain pipe inside an outside wall cavity? Perhaps, it really depends on how the building is constructed, not just how the wall is constucted. What structural members the drain and vent pipe must penatrate etc.
Any difference whether I use PVC or copper? Yes there are difference between PVC, ABS, Copper, and Cast Iron and people will argue over which is best until the cows come home.
Would freezing be a problem in winter? It could be. Particularly where water sits in the trap. I am not familiar with cold climate plumbing but there are also problems of small vent terminals freezing.
I am not a fan of Pex. You can call me old fashioned but I do not care to argue about it.