Short answers, pulled from the story.
Roman antiquarian Marcus Terentius Varro placed the city's foundation to 753 BC. The story of the founding dates at least to the 3rd century BC. Archaeological evidence reveals the first large-scale organization in the valley between the Capitoline and Palatine Hills in the middle of the 8th century BC.
Romulus and Remus were the twin brothers said to be the offspring of Mars and a princess of Alba Longa. Romulus killed Remus during a dispute and became the sole founder of the city. Their initial settlement on the Palatine Hill was later known as Roma quadrata.
The Roman Republic was established in 509 BC when the last of the seven kings of Rome, Tarquin the Proud, was deposed. A system based on annually elected magistrates and various representative assemblies was established. The most important magistrates were the two consuls who exercised executive authority.
In 27 BC at the age of 36, Octavian took the name Augustus and became the sole Roman leader. That event is usually taken by historians as the beginning of Roman Empire. Augustus assumed absolute powers while the government remained officially republican.
Nero is widely known as the first persecutor of Christians and for the Great Fire of Rome. Nero ruled from 54 AD until 68 AD when he killed himself. The fire was rumoured to have been started by the emperor himself.
Under Trajan the Roman Empire reached the peak of its territorial expansion. Trajan fought three Dacian wars and captured Northern Mesopotamian cities in 115 AD. He died in 117 AD after withdrawing due to health issues.