make sure you don't have an abcess/root canal infection. Get a second opinion and ask them to do an X-ray of the problem area to rule out the infection. The symptoms you are describing--throbbing pain, pain going into your brain, vicoding not helping, crown rising after root canal/initial crown installation--all seem to point in that direction.
I have had good experience with endodontists doign root canals and bad experience with general dentists doing them.
i couldnt even finish reading this i am so appalled!! they should never charge you for looking at a tooth they didnt take care of correctly. that crown should have never ever been placed with your discomfort. that $$ is on them. you might just have a 3rd nerve they cant see, its not uncommon to have a "hidden" canal. you need a specialist. they willl have to go into the crown which in my opinion should be remade after at their charge. this dentist needs to do something to make it right. a dr that EVER tells you its your fault , in my opinion, is unprofessional and is covering up for their own mistake they know they made. i hope you get something done soon to alleviate your pain.