Make the cake: Preheat the oven to 350 degrees F. Butter two 9-inch-round cake pans and line the bottoms with parchment paper; butter the parchment. Whisk the flour, cocoa powder, baking soda and salt in a bowl.
Beat the granulated sugar, 12 tablespoons butter and the vegetable oil in a stand mixer fitted with the paddle attachment on medium-high speed until fluffy, about 4 minutes. Beat in the eggs one at a time. Beat in the food coloring, vinegar and vanilla. Reduce the mixer speed to low; add the flour mixture in 3 batches, alternating with the buttermilk, beginning and ending with the flour, until just combined.
Divide the batter between the prepared pans. Bake until a toothpick inserted into the center comes out clean, 35 to 40 minutes. Let cool 10 minutes on racks, then turn the cakes out onto the racks to cool completely. Using a long serrated knife, carefully slice each cake in half horizontally to make 2 even layers.
Make the frosting: Beat the cream cheese and butter in a stand mixer with the paddle attachment on medium-high speed until fluffy. Add the confectioners' sugar, lemon juice, vanilla and salt; beat until smooth.
Put 1 cake layer on a platter; spread 1 1/4 cups frosting on top. Repeat to make 4 layers, ending with the cake. Cover the top and sides with a thin layer of frosting (this is the "crumb coat"; it doesn't have to be perfect). Chill 15 minutes, then cover with the remaining frosting.