That was, of course, when the Republic fell, but the Empire lasted another 500 or 1500 years, depending on how you figure it.
The Republic, BTW, was not murdered. It committed suicide. The last 100 years of its history it was ruled by aristocratic warlords who used the resources of the State to wage war against each other for control of the State.
The later Republic resembles an empire run by Mafia families more than anything else.
You have that correct. The Eastern Roman Empire lasted for another 1,000 years because they had a long string of good heads of state. The Western Empire had a long string of incompetents.