According to mythology, Romulus and Remus are two brothers who are raised by wolves in the wild. When they return to human society, they compete to found a city, and in the protracted disagreement, Romulus kills Remus and founds the city of Rome.

The story of Achilles, told famously in Homer's Odyssey and The Iliad, tells the tale of the Greek invasion of Troy and recounts the heroics of the Ancient Greek mythological hero.

As the question explains the Roman pantheon of Gods was based off of the Greek pantheon, but with different names. The head of the gods in Greece is Zeus, but in Roman religious practice his name is Jupiter. The only one of these which is incorrectly matched is Mercury and Athena. Mercury is the Roman equivalent of the Greek god of messengers called Hermes and Athena is the Greek version of the Roman god Minerva. 

In Ancient Greek mythology the twelve primary gods and godesses live atop Mount Olympus. Mount Olympus is the highest mountain in Greece and so a logical choice for a home for Gods in a religion where the gods routinely come down and intefere in the lives of humans. 

The two famous pieces of ancient Greek literature, The Odyssey and The Iliad, are a part of an oral tradition attributed to Homer.

Prior to the Fall of Rome, the Roman Empire had fractured into a Western Roman Empire and an Eastern Roman Empire around 280 CE. The Western Roman Empire crumbled under the difficulties of administering such a diverse empire and was practically defeated in the fifth century after being subjected to a wave of barbarian attacks on Rome, most notably attacks by the Visigoths and the Vandals (from whom we get the word "vandalism"). The Eastern Roman Empire would be persevered for another thousand years and has come to be called the Byzantine Empire by posterity.

From the time period mentioned, you can infer that this question is primarily referring to the downfall of the Roman Empire, which occurred (depending on who you ask) from about 300 CE to 550 CE. What most historians agree upon is that by 475 CE, the Roman political, military, and economic institutions were weakened beyond repair and much of the Western Roman Empire had been conquered by barbarians. The capital city of Rome was sacked numerous times during this time period, first in 410 CE by Alaric the Visigoth, then in 455 CE by the Vandals, then in 546 CE by the Ostrogoths.

The Punic Wars were fought between Rome and Carthage in the Third Century BCE. The wars resulted in victory for the Roman Empire and established Rome as the dominant power in the Mediterranean for the next several hundred years.

The Roman Empire lasted for almost a thousand years as the preemiment military and political power in Europe. At its height, the Roman Empire stretched from Scotland to Syria, Spain to Eastern Europe, and throughout North Africa; however, from about the fourth century CE onwards, Roman power was in decline. There are a myriad of reasons why the Roman Empire fell, and each historian has his or her own theory, but the most important reasons are as follows: barbarian invasions, as Germanic tribes sacked Rome and other imperial cities; economic stagnation, as Romans became overreliant on slave labor; weak leadership from a series of poor and ineffectual emperors; the rise of Christianity eroding traditional Roman values and systems of power; and overexpansion, as the empire simply became too vast to effectively maintain and police.

The Fall of the Roman Empire occurred in the 5th Century and was a protracted event. Rome had been declining in power for several decades before the city of Rome finally fell completely. It had suffered numerous sacks in the early years of the 5th Century - from the Visigoths and the Vandals. Interestingly our English word “vandalism” comes from this same barbarian tribe who greatly contributed to the destruction of Rome. A tangible example of the constant and continuing influence that the Roman Empire, and the Latin language, have on our lives.