Make the crust: Preheat the oven to 350 degrees F. Mix the cookie crumbs, melted butter, sugar and salt in a bowl. Press into the bottom and 1 inch up the sides of a 9-inch springform pan. Put the pan on a baking sheet and bake until set, about 10 minutes. Let cool completely.
Make the filling: Reduce the oven temperature to 325 degrees F. Beat the cream cheese, sugar, lemon juice and vanilla in a stand mixer fitted with the paddle attachment until smooth, 4 to 5 minutes. Add the flour, then beat in the eggs one at a time. Transfer 2 cups batter to a bowl; stir in the cocoa powder and food coloring. Pour the red batter into the crust, then pour the white batter on top. Using a spoon, pull some of the red batter up from the bottom of the cake and swirl. Bake until the edges are set but the center is still wobbly, about 1 hour, 20 minutes.
Turn off the oven but keep the cake inside to cool, 20 minutes. Run a knife around the edge of the pan (don't remove the springform side), then transfer to a rack to cool. Refrigerate at least 4 hours before slicing.