1. In a fine strainer, rinse rice under cold running water until water drains clear, 5 to 10 minutes. Put rice in a 2- to 3-quart pan and add 1 1/2 cups water. Bring to a boil over high heat, then lower heat to maintain a simmer. Cover and cook until water is absorbed, 10 to 15 minutes.
2. Meanwhile, in a small bowl, stir 1/4 cup rice vinegar, the sugar, and salt until sugar and salt are dissolved.
3. Spread rice out in a shallow 12- by 16-inch baking pan and slowly pour vinegar mixture over it, turning rice gently and fanning it until it comes to warm room temperature, about 10 minutes.
4. Mix remaining 1 tablespoon rice vinegar with 1 cup water. Pour into a small bowl to moisten your hands (so rice doesn't stick) while rolling sushi.
5. Place a sheet of toasted nori, shiny side down, on a bamboo rolling mat (with slats running horizontal to you) or on a piece of plastic wrap (about 10 by 12 in.), aligning long side of nori with the bottom edge of mat. Dip your hands in vinegar-water mixture and scatter 2/3 cup rice over nori; pat into an even layer, spreading out to the sides and bottom of nori, but leaving a 2-inch-wide strip bare along the top edge.
6. With your finger, spread a thin stripe of wasabi (about 1 teaspoon) horizontally across center of rice. Arrange four avocado slices along wasabi stripe (it's okay if the slices overlap or if some ends stick out), then arrange about 2 tablespoons of the crab on top of the avocado.
7. Moisten top edge of nori with vinegar-water mixture. Holding fillings down with your fingers, lift edge of mat closest to you with your thumbs and roll it over so that the near edge of nori meets the far edge of rice bordering the 2-inch-wide bare strip of nori at the top. Then roll bottom edge over to seal, and briefly press mat around roll.
8. Remove roll from mat. If desired, trim off any filling sticking out. With a sharp knife, cut roll into six or eight equal pieces, rinsing knife in water between cuts.
9. Repeat process with remaining nori, rice, and fillings. Serve with soy sauce, pickled ginger, and remaining wasabi.