Did they ever speak Latin in Rome or Greece?

I know Latin has roots in many languages, but I'm not sure about either of these. I need to know for a class, links would be great!
10 points for the most imformative answer!
Also if Latin did have roots in Greece than did is there Latin roots in the names of stuff in greek mythology (Perseus, Zeus, Pegasys, Medusa etc.)?

Byzantine Empire was said to be the legacy of the Roman Empire but why the people did not continue to speak Latin but instead spoke Greek? Was it because they moved to the modern Turkey, which is near to Greece.