Is there a way to inject medication into the gout joint to dissolve the crystals away?

Not at this time. Right now the only joint injectable medication will be a steroid injection to settle the inflammation down but in reality the crystals will not be disolved.
No. A similar medication called Krystexxa (pegloticase) can be infused through an I.V., but the vast majority of gout sufferers can be treated with anti-inflammatories or Colchicine for flares, and Allopurinol or Febuxostat to reduce uric acid.