Make the pork and beans: Place beans in a pot; add water to cover by 3 inches. Bring to a boil, and cook for 2 minutes. Remove from heat. Let stand, covered, for 1 hour. Drain.
Meanwhile, preheat oven to 300 degrees. Cook bacon in a large Dutch oven over medium-low heat until fat is rendered and bacon is crisp, about 10 minutes. Transfer bacon to a plate using a slotted spoon. Season pork shoulder generously with salt and pepper, and sear, skin side down, in Dutch oven until golden, about 5 minutes. Flip, and repeat on opposite side.
Return bacon to Dutch oven, and add drained beans, brown sugar, molasses, ketchup, and 1 tablespoon salt. Add water to cover (about 3 cups). Cover, and bring to a simmer. Transfer to oven, and braise, basting every hour, until beans are thick and syrupy and pork is falling off the bone, about 6 hours.
Meanwhile, make the pickled vegetables: Combine onions and jalapenos in a bowl. Bring cider vinegar, granulated sugar, and 1 teaspoon salt to a simmer in a small saucepan. Cook until sugar dissolves, about 4 minutes. Pour pickling liquid over onions and jalapenos. Let cool.
Transfer pork shoulder to a cutting board. Stir white vinegar into beans. Shred pork using two forks, and return to pot.
Assemble the sandwiches: Serve pork and beans on buns with shredded cabbage and pickled vegetables.
Pork and beans can be refrigerated for up to 3 days. Rewarm over low heat before serving (you may need to add some water if mixture looks dry).