Preheat a grill or grill pan to medium-high. Bring a large pot of water to a boil, salt it, add the spaghetti and cook until al dente. Drain, reserving 1 cup pasta cooking water.
While the pasta is working, in a bowl, combine 1 cup EVOO, the garlic, parsley and mint. Spoon about one-third of the herb dressing into a large skillet and heat over medium-high heat. Add the zucchini, season with salt and pepper and cook until softened, 8 to 10 minutes. Add the reserved pasta cooking water and the pasta; toss. Sprinkle in the cheese and toss. Keep warm.
Meanwhile, score the fish skin with a sharp knife in a crosshatch pattern. Season with salt and pepper; drizzle EVOO on both sides to coat. Grill the fish for 3 minutes on each side.
Divide the fish and arugula among plates. Squeeze the half lemon on top. Season the remaining herb dressing with salt and pepper; drizzle over the fish and greens. Serve with the pasta and lemon wedges.