Do you think I should go with 2" lumber or 3/4" with foam board insulation inside? The 2" might be more structurally solid, but I am not sure about the insulating benefits of the foam board. Not sure if there is any noticable difference in r-value between the two options.
Not exactly sure how you'll do this but 3/4" with foam will be more than strong enough (dependent upon your joints) and the foam (I suspect) would be superior to just 2x4 lumber.