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 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.