In a 4- to 5-quart pan over high heat, bring 3 quarts water to a boil. Add rice noodles, stir to separate, and cook until tender to bite, 3 to 5 minutes. Drain, rinse with cold water until cool, and drain again. Using scissors, cut noodles into about 3-inch lengths.
Meanwhile, stem and seed bell pepper; cut lengthwise into 1/4-inch slices, then crosswise into 2-inch lengths. In a small bowl, stir together fish sauce, sweet chili sauce, and lime juice.
Mound rice noodles in the center of a large platter. Drizzle 1 tablespoon fish sauce mixture over noodles, then sprinkle with chopped peanuts. Mound bell pepper, shrimp, chicken, greens, cucumber slices, shredded carrots, mint, and cilantro around noodles.
Pour 1 inch hot tap water into a large, shallow bowl. Submerge rice-paper wrappers, one or two at a time, until flexible and tender, about 30 seconds. Carefully remove and let drain briefly; stack on a plate.
Offer platter of fillings for diners to assemble their own summer rolls, with remaining fish sauce mixture alongside for dipping. To roll, add desired fillings in a strip across center of a wrapper; fold in sides over filling, then roll up tightly from the edge closest to you, like a burrito.