Place the pork shoulder in the slow cooker and add the sliced onion, minced garlic, apple cider vinegar, smoked paprika, salt, and black pepper.
Cover and cook on low for 8 hours until the pork is very tender and pulls apart easily with a fork.
Remove the pork from the slow cooker and shred it with two forks. Drain most of the cooking liquid, then return the shredded pork to the cooker and stir in the barbecue sauce until evenly coated.
In a large bowl, combine the shredded cabbage and carrot. In a small bowl whisk together the mayonnaise, apple cider vinegar, sugar, and celery seed. Pour the dressing over the vegetables and toss until well coated.
Split the slider buns and fill each with a generous portion of the pulled pork. Top with a spoonful of the tangy coleslaw and serve right away.