Yes
No, they were bitter enemies.
Cleopatra died by suicide when Egypt was being invaded by Augustus Caesar.
The Roman emperor who ruled after Cleopatra was Octavian/Augustus.
The Roman emperor that ruled after Cleopatra was Octavian or Augustus Caesar as he was later named.
Augustus declared war on Mark Antony and Cleopatra VII. He went to Alexandria to make Cleopatra surrender the throne. She didn't want to so she committed succide.
No there are no one left from Cleopatra's family. Augustus Caesar killed her children.
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Augustus, who was known as Octavian at the time, had a conflict over power. The conflict was between Antony and Octavian, not so much Cleopatra. Cleopatra was simply a handy tool for Octavian to use to rally forces against Antony. Even though it seemed that Cleopatra had her own ambitions, they were secondary to the major conflict between the men, which was all about power and the sole rule of the Roman empire.
To side with Mark Antony against Augustus.
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No, absolutely not. They were enemies. Although Cleopatra tried to seduce Augustus/Octavian, by making him "understand" her actions, Octavian would have nothing to do with her and in fact, would not even look at her when they spoke.
Octavian, later known as Augustus, became the emperor of Egypt after Cleopatra. He defeated Cleopatra and her lover Mark Antony in the Battle of Actium in 31 BC, which marked the end of the Ptolemaic dynasty and the beginning of Roman rule in Egypt.