Short answers, pulled from the story.
The city of Rome emerged from a cluster of small villages on the Palatine Hill in 753 BC and immediately engaged in warfare with surrounding Latin cities and the Sabines. The legendary account describes Romulus fortifying this hilltop settlement and engaging in desperate struggles for survival against immediate neighbors.
Gallic tribes known as the Senones attacked the town of Clusium near Rome's sphere of influence and defeated a Roman army of approximately 15,000 troops at the Battle of the Allia around 390 BC or 387 BC. The Gauls under chieftain Brennus pursued fleeing soldiers back to Rome itself and partially sacked the town before being driven off or bought off.
The First Punic War began in 264 BC when settlements on Sicily appealed to Rome and Carthage for help with internal conflicts. Before this war there was no Roman navy to speak of as all previous wars had been fought on land in Italy until the new conflict forced Rome to quickly build a fleet and train sailors.
Hannibal achieved victories at Trebia Lake Trasimene where he ambushed an unsuspecting Roman army and Cannae considered one of the great masterpieces of tactical art during the Second Punic War which started when Hannibal attacked Saguntum a city with diplomatic ties to Rome. He raised an army in Iberia and famously crossed the Italian Alps with elephants to invade Italy.
Extensive campaigning abroad led soldiers to become increasingly loyal to their commanders rather than the state resulting from rewarding troops with plunder from campaigns. Between 135 BC and 71 BC three Servile Wars occurred against the Roman state and the third most serious may have involved revolution by 120,000 to 150,000 slaves.