just store them in a tin or under plastic wrap at room temperature. they'll be fine for several days, maybe even a week. i find that refrigerating cake tends to make it a bit tough.
Yes. Store in an airtight container that will not touch and therefore remove the frosting. They will last days, if you don't eat them by then. Refrigeration negatively impacts cake texture, even after the cake comes back to room temp. Of course, if refrigerated cake is all there is, I will eat it.