Make the strudel dough: In the bowl of a stand mixer fitted with the dough hook, combine the flour with the wine, oil, sugar, salt, and 1⁄4 cup water. On low speed, mix until the dough comes together, and then increase the speed to medium and knead until smooth and elastic, 8 to 10 minutes. Wrap the dough in plastic wrap and refrigerate for 1 hour or up to 24 hours.
Make the cheese filling: In a small bowl, soak the raisins with the rum until soft, 30 minutes. Stir in the cheese, 1⁄4 cup sugar, 1⁄4 tsp. salt, and vanilla into the raisins. Pour 1 tablespoon of the beaten egg into the cheese filling and stir to combine; reserve the remaining beaten egg for Step 9.
Make the poppy seed filling: In a small saucepan, combine the ground poppy seeds with the milk, 1⁄3 cup sugar, 1⁄4 tsp. salt, and 1⁄4 tsp. cinnamon and bring to a boil over medium-high heat. Cook, stirring, until the poppy seeds form a smooth paste and just begin sticking to the bottom of the pan, 3 to 4 minutes. Remove from the heat, scrape into a small bowl, and let cool completely.
Make the apple filling: In a small bowl, toss the apples with 1⁄4 cup sugar, lemon juice and zest, 1⁄4 tsp. salt, and 1⁄8 tsp. cinnamon. Let stand for at least 20 minutes before using.
Make the walnut filling: In a small skillet over medium heat, toast the walnuts until fragrant, about 6 minutes. Finely chop the walnuts and place in a small bowl. Stir in 1⁄4 cup sugar, 1⁄4 tsp. salt, and 1⁄8 tsp. cinnamon until evenly combined.
Assemble the cake: Line the bottom of an 8-inch round springform pan with parchment paper and then brush the bottom and side with some of the melted butter. Unwrap the strudel dough and divide into 9 pieces (1 piece should weigh 3 1⁄2 oz. and the remaining 8 should weigh 1 3⁄4 oz.). Using a rolling pin, flatten the larger, 3 1⁄2-oz. piece into a 12-inch circle. Line the baking pan with the dough circle, letting the excess hang over the edge of the pan.
Brush the dough with melted butter and then spread half of the poppy seed filling over the dough, smoothing the top. Flatten each of the eight 1 3⁄4-oz. pieces of dough into an 8-inch circle, and then place 1 circle over the poppy seed filling. Brush the circle with butter and then spread half of the apple filling over the top. Cover with a second dough circle, brush with butter, and spread half of the cheese filling over the top. Cover with a third dough circle, brush with butter, and sprinkle with half of the walnut filling. Repeat with 4 more dough circles and the remaining poppy seed, apple, cheese, and walnut fillings, brushing each dough circle with butter before adding the next filling.
Fold the overhanging dough back into the pan, over the walnut filling, and brush the dough with butter. Place the last dough circle over the walnut filling and press gently at the edge to adhere to the buttered dough. Brush the top of the pastry cake with the remaining butter and pierce the top with a paring knife in 6 evenly spaced places. Refrigerate the cake at least 4 hours, or overnight.
Bake the cake: Heat the oven to 350°. In a small bowl, stir the sour cream with 1 tablespoon of the reserved beaten egg until smooth; discard the remaining egg. Spread the sour cream evenly over the top of the pastry cake, leaving a 1⁄2-inch border, and cover with foil. Bake the pastry cake for 1 hour, and then remove the foil and bake until the edges are golden brown and an instant-read thermometer inserted in the middle of the cake reads 180°, about 30 minutes more.
Transfer the cake to a rack and let cool completely. Unmold the cake from the pan, transfer to a serving platter, and discard the parchment paper. Dust the top of the cake with confectioners' sugar before serving.