Preheat the oven to 325 degrees F; coat a 9-inch-round springform pan with cooking spray. Prepare the cake mix as directed, adding the cocoa powder and food coloring to the batter. Transfer to the prepared pan and bake until done, about 45 minutes.
Let the cake cool, then remove from the pan; level the domed top with a serrated knife. Use a 5-inch circle of parchment paper as a guide to cut a circle from the center of the cake (you won't need the small circle). Freeze the large cake ring.
Cut a 6-by-30-inch strip of parchment paper. Clean the springform pan, then butter it and line the side with the parchment as shown; the paper will extend above the rim of the pan so you can build a tall cake.
Let 1 pint sorbet soften slightly at room temperature. Spread in the prepared pan, then lay a piece of plastic wrap on top and use the bottom of a measuring cup to pack the sorbet into the pan in an even layer. Freeze until firm, at least 30 minutes.
Let 1 pint vanilla ice cream soften; spread in the pan, cover with plastic wrap and press evenly with the measuring cup. Freeze until firm, then repeat to make 1 more layer each of sorbet and ice cream (4 layers total). Freeze until firm.
Place the blue cake ring on top of the ice cream, pressing gently; return to the freezer while you prepare the next layer.
Cut off the top 1 1/2 inches of the remaining sorbet carton; remove the carton and wrap the sorbet loosely in plastic wrap. Press into a 5-inch round, then place the flattened sorbet into the hole in the cake, smoothing as needed. Freeze until firm.
Cut off the top 1 1/2 inches of the remaining ice cream carton (you won't need the bottom). Flatten as you did the sorbet; place on top of the raspberry layer. Repeat with the bottom section of sorbet. Freeze until firm.
Beat the heavy cream and confectioners' sugar with a mixer until stiff peaks form. Remove the springform ring and parchment collar from the cake. Cover the cake with the whipped cream. Freeze until ready to serve.