No mention is made of Cleopatra's three children by Marc Antony, twins Alexander Helios and Cleoptra Selene II (born 40 BC), and Ptolemy Philadelphus (born 36 BC). After Octavian conquered Egypt, they were sent to Rome, where they would eventually be raised by Octavia Minor, Octavian's sister and Marc Antony's wife. Marc Antony also had at least five children before he fell in love with Cleopatra, none of whom are mentioned.
Arsinoe was captured by the Romans, but not strangled by Cleopatra's guards in Egypt as shown. She was taken as a captive to Rome in 46 BC and remained alive, living in a Roman temple until 41 BC, when she was executed on Marc Antony's orders.
Octavian was actually not in Rome when Caesar was murdered. He only returned to Rome when told of Caesar's death.
Caesarion was 17 when Octavian invaded in 30 BC, not a child as is shown.
Cleopatra mentions the Octavian is not Caesar's 'flesh and blood'. Octavian was in fact, Caesar's grandnephew, the son of Caesar's niece Atia (though Caesar introduces him as simply his 'nephew', whether this is a goof or just simplification on Caesar's part is unclear)
During the battle within the walls of Alexandria, Cleopatra is running to save the burning library and is chased down by a soldier on a horse. Cleopatra uses her own spear to flip him into a pillar. When he hits the base of the pillar, it moves, revealing that it is a loosely attached prop.
After Cleopatra died, you could still see her tummy moving.
The legionaries are dressed like the Imperial Legions with lorica segmentata armour (plate in sheets held together by joints) and not as the post marian reform units they really were at that time that would wear a sort of chainmail armour or "Lorica Hamata". It is contested by some historians even that the Lorica Segmentata never reached widespread use and even maybe only functioned as a ceremonial type of armour.
In any case the first known (minor)a ppearance of this armour is in 19 BC, some time after the events portrayed in the movie.