Assuming that, since the jaw disease is taxa-specific, that would mean that the afflicted taxa would share the same segment of the genome that causes the animal to get sick. Does the discovery of this jaw disease in T. rex give us a clue as to what a segment of a T. rex genome looked like? Would this mean that that gene (whereever it is) was shared between birds and T. rex? How many genetic diseases to living birds have? Could they provide us any clues as to the gene structure of tyrannosaurids?
Being an infection with *Trichomonas*, the disease in question is not genetic. Your other questions, however, stand: what is it that makes birds -- or rather coelurosaurs -- susceptible to *Trichomonas* infections that's missing in other vertebrates?