Place the rice in a bowl and wash with cold water until the water runs clear. Drain and place in a 3-quart saucepan with the 3 3/4 cups cold water. Bring to a boil. Cook, uncovered, stirring occasionally, until all the surface water is absorbed, about 3 minutes. Cover, reduce the heat to low and cook, without stirring, until tender, about 20 minutes. Set aside for 10 minutes. Alternatively, cook the rice in a rice cooker according to the manufacturer's instructions.
In a small bowl, combine the 1/3 cup water and 2 tsp. vinegar. In another small bowl, combine the wasabi powder and 1 1/2 Tbs. water, stir to form a smooth paste and let stand for 10 minutes. Divide in half and set half aside. In a small saucepan over low heat, combine the 1/2 cup vinegar, the sugar and salt and cook, stirring occasionally, until the sugar and salt are dissolved, about 3 minutes. Set aside to cool.
Transfer the hot rice to a large bowl. Drizzle with two-thirds of the vinegar-sugar mixture and gently fold into the rice. Add only as much as the rice will absorb without becoming mushy. Cover with a damp kitchen towel.
Place a bamboo sushi mat on a work surface with the bamboo strips running horizontally. Place 1 nori sheet aligned horizontally on the mat, shiny side down, with the edge nearest you. Dip your hands into the vinegar-water mixture and spread about 2 cups of the rice in an even layer over the nori sheet, leaving the top one-fourth uncovered. Smear a thin strip of wasabi horizontally across the middle. Sprinkle sesame seeds over the wasabi, followed by a few cucumber strips, an even row of avocado slices, and one-sixth of the crabmeat. Lift the nearest edge of mat, nori and rice over the filling to seal it inside. Continue to lift and press on the mat to form a snug cylinder about 2 inches wide. Dipping a sharp knife in water before each cut, cut the roll in half. Cut each half into 4 equal pieces. Repeat with the remaining 5 nori sheets.
Serve the sushi with the reserved wasabi, the soy sauce and the pickled ginger. Provide small dishes for mixing a small amount of wasabi with soy sauce for use as a dipping sauce.