A 10-inch cake is WAY too big for 2 adults and 2 kids. You would have leftovers with even a 6-inch cake if it's a fancy fondant-covered kind like you get at the Deluxe resorts. I'd at least switch to an 8-inch, but you really don't need more than a 6-inch. And only a 6-inch is going to fit in the mini fridge. I definitely recommend refrigerating the leftovers. I usually bring Tupperware to put them in, since those fit better in mini fridges than the cake box and will keep the cake fresher. Buttercream-filled cakes taste better longer than mousse-filled cakes (and the mousse ones definitely require refrigeration, whereas you can get away with leaving buttercream-filled cakes out).