For the tartar sauce: Whisk together the mayonnaise, dill pickle and brine, Worcestershire, mustard, capers and brine, hot sauce, garlic, tarragon, basil stems and lime zest and juice in a medium bowl. Taste and season with salt and pepper. Set aside some in a ramekin.
For the batter: Whisk together the flour, beer, sparkling water, vodka, paprika and baking soda in a medium bowl.
For the frying: Heat the oil in a large heavy-bottomed pot fitted with a thermometer until it reaches 350 degrees F. Line a sheet tray with a cooling rack and set aside.
Place each whole squid on a flat surface and run your knife gently down the length of the outside. Peel off the outer layer of skin and remove the piece of cartilage and anything inside the tube of the squid. Cut the tentacles from the body (below the eyes) and remove the sharp beak (which feels like a small gumball) in the center of each set of tentacles. Cut into 1/8-inch rounds. Dry the rings and tentacles on a sheet tray lined with paper towels.
Fry the basil leaves until crispy, 30 seconds to 1 minute. Remove from the oil, letting the excess drain off, and place on the cooling rack. Season with salt immediately. Fry the lemon slices until crispy, 1 to 2 minutes. Drain the excess oil, then place on paper towels and season immediately. Transfer the lemons and basil to a bowl. Replace the paper towels for the next batch.
Place a hand strainer in the batter (to easily find pieces) and sprinkle the squid with salt. Starting with the tentacles, fully coat in the batter, then drop into the hot oil. Fry until crispy, 2 to 3 minutes. Drain and salt. Repeat with the rings. Bring the oil back to 350 degrees F and dip the shrimp in the batter. Drop in small batches into the hot oil. Fry, turning on both sides with a slotted metal spoon until crispy, 1 to 2 minutes. Drain and salt.
Transfer the tartar sauce ramekin to a medium serving bowl and arrange the fried lemons, basil leaves, squid and shrimp around it. Serve immediately.