Prepare grill for direct heat and preheat to medium-high.
Combine olive oil, vinegar, soy sauce, sesame oil, sriracha sauce, garlic and ginger pastes in a small jar. Shake until fully combined. Pour about ¼ of the dressing into a separate bowl.
Pat the shrimp, scallops, and squid dry with paper towels. Brush the seafood with the dressing you poured out. Discard after brushing. Put the shrimp and scallops into a grill basket that has been sprayed with oil.
Oil the grates of the grill well. Place the grill basket on the grill and the squid bodies directly on the grates. (Can also be threaded on skewers and grilled if no grill pan.) Grill the shrimp and scallops until opaque, about 1½ - 2 minutes per side. Remove from heat and place in a bowl. Grill the squids until charred and cooked through, about 3 minutes per side. Let cool enough to handle then cut into rings and add to the shrimp and scallops. Pour on some of the dressing and toss well. Season to taste with salt and pepper.
Put the salad greens in a salad bowl, toss with some dressing and season to taste with salt and pepper. Add the seafood, toss gently, and serve.