Is there an example of the Science or Medical Departments studying the information recorded during the beaming process to diagnose health issues among those that used the Transporter system?
Yes, but onscreen it didn't happen very often. From the Transporter article on Memory Alpha:
Transporter traces were also used as a medical tool, to help in spotting anomalies at the molecular level. When comparing the transporter ID traces of Deanna Troi, Data and Miles O'Brien before and after they were taken over by Ux-Mal criminals, Doctor Crusher was able to detect that their nervous systems were generating high levels of synaptic and anionic energy. (TNG: "Power Play") Another example of such an application was in 2373, when The Doctor used Harry Kim's transporter trace records to determine when he had been infected with Taresian DNA. (VOY: "Favorite Son")
Biofilters were part of the transporter system, so that diseases and other known pathogens were removed from the transportee before they dematerialized, making such examinations usually unnecessary.