Cover a rimmed baking sheet with ice (to later cool down shrimp). In a large stockpot combine 12 cups water, carrot, celery, thyme, salt, and peppercorns. Cut one lemon in half; squeeze the juice into the water and add the rind. Cut remaining 1 1/2 lemons into small wedges for serving; set aside. Bring water to a boil. Add shrimp, cover; remove from heat. Let stand until shrimp are bright pink and curled; 3 to 5 minutes. Drain and transfer the shrimp to the prepared tray of ice to cool completely. Meanwhile, whisk together all cocktail sauce ingredients (ketchup, lemon juice, horseradish, salt, pepper, and hot sauce) until combined. Arrange shrimp around individual servings of cocktail sauce, add a fresh lemon wedge, and serve.