Whisk together the ketchup, horseradish sauce, hoisin sauce, paprika, and cayenne pepper to make the cocktail sauce. Chill until ready to serve.
In a separate bowl, stir together the mayonnaise, soy sauce, sriracha, and lemon juice. Chill until ready to serve.
Preheat a deep fryer or pan filled with two inches of vegetable oil to 325°F. Use a sharp knife to cut the cleaned squid into rings.
Mix together the tempura mix and ice water. The batter will be a bit chunky, and that's okay. Dip each squid ring into the batter, then carefully toss into the fryer for 1-2 minutes or until the squid turns golden brown.
Transfer the fried squid rings from the fryer to a plate lined with paper towels to remove any excess oil.