Preheat the oven to 400 degrees F. Heat the rice as the label directs; keep warm. Chop 1 to 2 tablespoons jalapeños; set aside. Combine the beans, corn and salsa in a large skillet over medium-high heat. Bring to a simmer and cook until thickened, 2 to 3 minutes. Stir in the shrimp and simmer until cooked through, 5 to 7 minutes. Stir in 1/4 cup cilantro, the chopped jalapenos and brine.
Mash the avocado with the lime juice and remaining 1/4 cup cilantro in a small bowl. Season with salt and pepper.
Warm the tortillas in the microwave until softened, 30 seconds to 1 minute. Spread the avocado mixture in the center of each tortilla. Top with some of the shrimp-bean mixture, then top with the rice. Fold in the sides of the tortillas and roll up.
Place the burritos seam-side down on a baking sheet and bake until warmed through, about 5 minutes. Serve with sour cream, lime wedges and pickled jalapeno slices.