Make cake: Preheat oven to 325°F. Mist a 12-by-17-inch rimmed baking sheet with cooking spray; line with parchment. Sift cocoa and flour together in a small bowl.
Beat yolks and zest until thick, 8 minutes. In a separate bowl, beat egg whites until fluffy. Add sugar; beat until glossy and stiff. Fold 1/4 of whites into yolks, then fold in remaining whites. Fold in juice and flour mixture. Do not overmix. Pour batter into pan; spread. Bake until cake springs back when touched, 12 minutes.
Sift enough cocoa onto a large kitchen towel to cover it. Invert warm cake onto towel. Carefully peel off parchment. Starting at long end and using towel as a guide, roll up cake and towel. Transfer roll, seam side down, to wire rack.
Make filling: Beat cream cheese, butter, juice and zest until creamy. Gradually beat in sugar and salt. Cover; chill until firm but spreadable, 40 minutes.
Make frosting: Melt chocolate; cool to room temperature. Mix cocoa, juice and 1 Tbsp. hot water. Beat butter, sugar and salt until fluffy. Mix in chocolate, then cocoa mixture.
Assemble cake: Unroll cake; remove towel. Reserve 2 T bsp. filling. Spread remaining filling over cake. Reroll cake. Cover with towel and chill, seam side down, until filling is set, about 30 minutes.
Cut cake on a diagonal; removing a 3-inch piece from each end for branches. Spread 1 T bsp. of filling on each piece; press against long side of cake at opposite ends. Frost cake. Draw lines with a fork to resemble bark. Draw circles on branches to look like rings. Chill.
Remove cake from refrigerator 1 hour before serving. Sift confectioners' sugar over cake, if desired.