Combine 5 tbsp. vinegar, salt, and sugar in a bowl; set aside. Put rice into a fine-mesh strainer. Lower strainer into a bowl of cold water. Gently rub and toss rice in water. When water is milky white, remove strainer; drain bowl and refill. Repeat process 2–3 more times until water is almost clear. Drain rice; transfer rice to a medium pot, add 2 1⁄4 cups water, and let sit for 20 minutes. Cook over medium heat until water is nearly absorbed, about 15 minutes. Reduce heat to low, cover, and cook until plump, about 15 minutes. Uncover pot very quickly: rice should be completely transparent. If you see any dry spots, sprinkle a little warm water over them and cook for 1–2 minutes more. Remove pot from heat. Set aside, covered, to let rest for 5 minutes.
Transfer rice to a bowl; use a wooden paddle to gently break it up. Add reserved vinegar mixture; break up clumps, turning rice over. Push rice toward one side of bowl. Insert paddle into rice and rapidly move it back and forth, breaking up clumps and pushing a portion of rice toward opposite side of bowl. The vinegar mixture should be evenly absorbed. Using a folded newspaper, fan rice for 30 seconds. Cover with a moist kitchen towel; let cool slightly.
Combine mayonnaise and chile sauce in a bowl; set aside. Combine remaining vinegar and 1 cup cold water in a bowl; set aside. Wrap a bamboo sushi rolling mat tightly with plastic wrap. Top mat with 1 piece nori, shiny side down, with the long sides parallel to you. Moisten hands with vinegar water and pick up a scant cup of rice, forming it into a ball without squeezing. Place rice on nori and spread it to cover the nori, except for a 1⁄2"-wide band at the top. Sprinkle surface of rice evenly with 1 tbsp. of sesame seeds to form an outer coating. Flip nori over, placing the uncovered band of nori closest to you. Smear about 2 tsp. of the mayonnaise mixture across nori, leaving the uncovered band dry. Arrange crabmeat along center of nori and top with 2 sticks cucumber and 2 slices avocado, to form a horizontal line. To roll, pull the band of uncovered nori over the filling; lift up bamboo mat, fold it over filling, and roll forward. Hold the roll with one hand (over the mat) and pull back on the other edge of the mat with the other hand to tighten the roll. Continue to roll tightly. Remove mat and drape over roll; press it around the roll to firm it up. Cut roll into 8 pieces and transfer to a platter. Repeat process with remaining ingredients. Serve with wasabi, pickled ginger, and soy sauce, if you like.