Make the court-bouillon: Prepare a large ice-water bath. Fill a 3-gallon pot with cold water. Add wine, salt, lemons, paprika, coriander seeds, celery seed, and bay leaves. Bring to a boil.
Cook seafood: Add lobsters head first to court-bouillon. Cover; return to a boil. Cook 10 minutes. Using tongs, transfer lobsters to ice-water bath. Add crab legs to court-bouillon. Return to a boil; cook 8 minutes. Transfer lobsters to a plate; immediately add crab legs to ice-water bath, adding more ice as needed. Add shrimp to court-bouillon. Return to a boil; cook until shrimp are pink and opaque, about 3 minutes. Add shrimp to ice-water bath; let crab legs and shrimp cool completely. Transfer to a plate.
Separate tails, claws, and knuckles from lobsters; using kitchen shears or a knife, split tails lengthwise through shells, then halve crosswise. Crack claws and knuckles so that meat can be easily removed. Separate crab legs at knuckles. Using kitchen shears, cut away a piece of shell so meat can be easily removed. Peel and devein shrimp, leaving tails intact. Transfer lobster pieces, crab legs, and shrimp to a rimmed baking sheet; wrap tightly in plastic. Refrigerate until ready to serve, or overnight.
Assemble the platter: Just before serving, fill a 3-tier glass serving stand with crushed ice. Shuck the oysters and clams; set flat on bottom tier so that juices do not spill. Arrange lobsters and shrimp on middle tier. Put caviar into a small bowl, and set in center of top tier; arrange crab legs around caviar. Serve with toast points, the cocktail sauce, saffron mayonnaise, mignonette sauce, and lemon wedges.