So the evacuated layer does nothing for the specific heat capacity. The radiation would come from anything hot. If it is hot outside, then the radiation is trying to get in. If it is hot inside, then the radiation is trying to get out. This layer will not decrease radiation.
Conduction is the passing of heat through a medium.
The only way heat can pass through a vacuum is via radiation. You may have heard me say in class that a layer of air is a good insulator, well, vacuum is even better.