All three of these Jewish groups began in the Middle East, according to the map provided. You can see that they originate from areas near Egypt and Arabia. But over time, the Jews spread to different parts of the world such as the Roman Jews going to Italy whilst the Exile going to Mesopotamia and Egypt.
Unfortunately, I can't answer that last question as I don't know much information about the Torah or what role it played during that time period. I hope this helps you someway regardless! Good luck! :)
<The Torah has always been the general law of the Jews, being the moral and civil base of the whole development of that society, which has always remained united around these principles. The Jewish beginning came near the lands of Jerusalem, when they were expelled from there by the Romans came to occupy the regions of Turkey, Egypt, Greece, and south of the Italian peninsula. After that period it occupied the regions to the northwest of Africa, as well as the south of Spain, France and, at the end of the Roman period already it was possible to find Jews to the north of France and even in Great Britain.>
