Place rice in a wire-mesh strainer; rinse with cold water.
Sauté garlic and shallots in hot oil in a Dutch oven over medium-high heat 2 minutes or until tender. Add 6 cups water, rice, and salt; bring to a boil. Cover, reduce heat, and simmer 20 minutes or until water is absorbed and rice is tender. Remove from heat; let stand, covered, 10 minutes. Fluff with a fork.
Whisk together rice wine vinegar and next 4 ingredients; pour over rice mixture, tossing gently.
Cut green onions in 1-inch lengths; cut into thin strips. Add onions, carrot, and bell pepper to rice mixture; toss gently. Cover and chill up to 3 days.