In a pot of boiling salted water, cook the spaghetti until tender. Drain and rinse under cold water until cooled. Drain well.
In a blender, puree the peanut butter with 6 tablespoons of the vinegar, 3 tablespoons of the sugar, soy sauce, water, sesame oil, crushed red pepper, ginger, and garlic. Transfer ½ cup of the peanut dressing to a bowl and toss with the noodles.
In another bowl, toss the celery with the cilantro and the remaining 2 tablespoons of vinegar and 1 teaspoon of sugar.
Transfer the noodles to bowls and drizzle with the remaining peanut dressing. Top with the celery and serve with lime wedges.