If someone was urinating, and they were to hypothetically suddenly "fall dead" (the brain is unconscious and all muscles / organs are unfunctional), would they continue urinating or would it stop?
Yes. Moreover, even if they were not urinating at the time of death, they likely would at death, assuming adequate filling of the urinary bladder. There are two sphincters that close off the urethra. One is under control of the autonomic nervous system and the other is voluntarily controlled (somatic). At death, both sphincters will relax, releasing the bladders contents.