Preheat oven to 350°F. Line bottom and sides of a 13- x 9-inch baking pan with parchment paper, leaving a 2-inch overhang on 2 opposite sides; set aside.

Whisk together flour, salt, and baking soda in a medium bowl; set aside.

Beat together sugar and butter in a large bowl with an electric mixer on medium until light and fluffy, 3 to 4 minutes. With mixer on low, add eggs, 1 at a time, beating until just combined after each addition. Add vanilla, lime zest, and lemon zest; beat on medium until combined, about 10 seconds.

Whisk together soft drink, 3 tablespoons lime juice and 2 tablespoons lemon juice in a medium bowl. With mixer on low, gradually add flour mixture alternately with soda mixture, beginning and ending with flour mixture, beating until just blended after each addition.

Spread batter into prepared baking pan. Bake until a wooden pick inserted into center of cake comes out clean, 30 to 35 minutes. Let cake cool in pan on a wire rack for 10 minutes. Using excess parchment overhang as handles, carefully lift cake from pan; let cool completely on a wire rack, about 1 hour. Place cooled cake on a platter, and arrange so the long side is horizontal; set aside.

Make the frosting: Beat butter and salt in a large bowl with an electric mixer on medium until creamy, about 30 seconds. With mixer on low speed, gradually add 3 cups powdered sugar, beating for 1 to 2 minutes total, stopping to scrape down sides of bowl as needed. Add 1 tablespoon lemon juice and 1 tablespoon lime juice; beat on medium until smooth, about 15 seconds. With mixer on low speed, gradually add remaining 1 cup powdered sugar. Increase mixer speed to medium; beat until light and fluffy, about 1 minute. Spread frosting evenly over top and sides of cooled cake.

Arrange blueberries in neat rows in top left corner of cake to form the stars of the flag. Arrange strawberry slices horizontally in 5 rows to form the stripes. Slice and serve.