Plot hole: Death row prisoners cannot donate organs. No doctor participates in an execution and after the lethal injection the medical examiner must wait at least fifteen minutes to test for cardiac activity before pronouncing the prisoner dead. This is enough time for hypoxia to render the heart, lungs and kidneys useless for transplant. Apart from this, organs intended for transplant must be removed within minutes of the death of the donor. Prisoners are not executed in operating theatres.
Plot hole: Technically, Brian Babbage's plan would never have worked due to the fact that all outbound mail goes to a main post office from the local post office and then placed through scanners to determine the contents, which would have inevitably exposed the bomb inside.
Plot hole: Monk identifies the killer because of a shirt sleeve that gets ripped off... he noticed the button is sewed on in a unique manner with the threads parallel as opposed to the "standardized" crisscross pattern. The killer and Monk coincidentally use the same dry cleaner and Mrs. Ling sews buttons parallel because it's her style. Dialogue in the episode indicated she is the only one who sews buttons on like that. But I checked over 20 button up shirts here at my house and the ratio of crisscross to parallel button sewing methods is 50% of each - hardly some unique identification method.