i have heard of some issues numbing upper teeth but never lowers unless there is infection present. the bubble is an abscess and until the infection is cured you might remain unnumbable. since the first course of antibiotics obviously didnt work you need a different kind.
What you describe is not that hard to take out. Many general dentists get very good at extractions, and any oral surgeon worth his license can remove that tooth in 15 minutes or less.
Well There's only a tiny little bit of the tooth above the gums. They can still take it out normally? The part that's still above the gums is a tiny part that touches the tooth in front of it. When the dentist was trying to get it out he had ripped off part of the enamel. It just kind of started crumbling. Are you sure they can still take it out normally if I can find a dentist who can numb me?
Go to another dentist, or hopefully an oral surgeon. It can be removed normally, you just need to find someone who knows how to get you numb, because your current dentist obviously does not.