Finely grate enough zest from limes to measure 1 tablespoon. Cut limes crosswise in half; squeeze juice from limes. Place zest and juice in a blender; add cilantro, sugar, red chile pepper, garlic, salt and pepper. While processing, gradually add oil in a steady stream.
In a large bowl, combine cabbages, cucumber, edamame, avocados, carrots and green onions. Drizzle with dressing; toss to coat.