Everything the team says about GVHD (graft-versus-host disease) has nothing to do with reality. GVHD is the most common complication after transplantation and with 4 out of 6 match can't be as dangerous as the characters say.
Activating the cardiopulmonary bypass machine during open heart surgery does not immediately stop the heart as shown. The heart muscle is self polarizing, and will continue to beat of its own accord even if removed from the body. For this reason the heart must be stopped chemically, most commonly by using a solution of potassium.