Bring first 3 ingredients and 4 1/2 cups water to a boil in a medium saucepan over high heat. Stir in rice. Cover, reduce heat to medium-low, and simmer 15 to 20 minutes or until rice is tender and liquid is absorbed. Stir in rice wine using a fork.
If frozen, thaw shrimp according to package directions. Peel shrimp; devein, if desired.
Cook shrimp in boiling water to cover 2 to 3 minutes or just until shrimp turn pink. Drain, and let cool 15 minutes. Coarsely chop shrimp.
Cut bell peppers into 1/4- to 1/2-inch-wide strips.
Coat a griddle with cooking spray, and heat over medium heat. Pour a thin layer of beaten eggs onto griddle to make a thin omelet. Cook, without stirring, 1 minute or until egg is cooked and dry. Transfer omelet to a clean kitchen towel. Repeat procedure with remaining eggs. (Do not stack or overlap omelets on towel.)
Arrange omelets in a single layer in a plastic wrap-lined 13- x 9-inch pan, overlapping edges and piecing together as needed to cover bottom of pan. Sprinkle capers and toasted sesame seeds over egg layer. Arrange salmon over half of mixture in pan; arrange shrimp over other half. Using wet hands, top with 2 1/2 cups rice mixture, pressing into an even layer and packing down firmly.
Arrange spinach over rice, overlapping edges. Top with green onions. Using wet hands, top with 2 1/2 cups rice mixture, pressing into an even layer and packing down firmly.
Remove and reserve 1 bell pepper strip. Arrange remaining bell pepper strips over rice. Top with avocado. Sprinkle with lime juice (about 1 Tbsp.). Using wet hands, top with remaining rice mixture, pressing into an even layer and packing down firmly. Cover with plastic wrap, and place a second 13- x 9-inch pan, right-side up, directly on filled pan, pressing down firmly on topmost pan to compact layers. Chill at least 2 hours or up to 8 hours.
Remove top pan; remove and discard plastic wrap. Invert sushi onto a cutting board, and remove and discard plastic wrap. Using a thin sharp knife, cut sushi into 1- to 1 1/2-inch pieces, cutting straight down through sushi. (Do not use a sawing motion when cutting.)
Cut reserved bell pepper strip into tiny diamond shapes or pieces; top each sushi bite with a bell pepper piece. Serve with desired toppings.