The Republic expanded so rapidly because of the Roman willingness to incorporate other Italian states/cultures into their own state apparatus.

Relatively early, Romans had a system of privilege sharing with allied groups that was in contrast to harsher system imposed by victors upon the conquered in the Ancient world. Romans would extend certain benefits of citizenship to conquered communities- a novel practice in the ancient Mediterranean. An extension of these latin rights (ius Latii) in addition to colonization projects allowed for integration of new areas into the Roman state apparatus and cultural sphere.

By the time the Romans came into conflict with Carthage, they already had the majority of the Italian penninsula to draw on as a population resource. Additionally, the Roman willingness to adopt useful artifacts/traditions from foreign cultures made them an extremely adaptable foe throughout most of its history (many great elements of the Roman military were adopted piecemeal throughout the Republican period).

These things explain (in part) why the Romans were able to build a larger and longer-lasting empire than any other Mediterranean society.

It is a very subjective question so there can never be a truly "correct" answer to this question, but I think that there are a few overarching factors to take into account:

• Military invention and perseverance: At the start of the Roman republic, there wasn't really anything particularly unique about the city-state; I doubt anyone at the time would have thought it would develop into what it eventually did. But one of the things that allowed Rome to conquer and expand so much was their openness to adapt their military tactics. Throughout Roman history we see various key reforms in the structure of their military. They started out with the classic Phalanx formation similar to tactics we know about in Greece, but eventually abandoned it in favor of the Legion. This was the first of many major changes to the structure of the army, probably the most famous of which are the Marian Reforms. This ability to adapt to changing times allowed its military to flourish and conquer such a huge empire. Additionally, while this slightly more subjective (but so is the question), the Roman ability to bounce back form huge losses as a key in their success. You mention in your post about Roman losses, and you are absolutely right, throughout Roman history, what is almost as amazing as their numerous victories are their numerous losses. The best example of this is probably the famous Battle of Cannae, where the Romans were crushed by Hannibal but eventually still won the war against the Carthaginians (http://www.unrv.com/empire/battle-of-cannae.php). Other famous losses as the Battle of the Teutoburg Forest point to the Roman ability to bounce back from major defeats, something relatively uncommon in ancient times.

• Leadership and Luck: I think that a huge factor in the rise of Rome was the superb leadership they had throughout their development, and their extreme luck when leadership fell into the wrong hands. During the days of the republic, the relative meritocracy of the Roman system helped produce great leaders that helped produce the Mediterranean hegemony that we associate with the empire. After the start of the empire, Rome definitely had its share of good and bad leaders, with a high correlation between the bad ones being born into the position and the good ones getting their through selection or seizing the moment. In your question you refer to the "zenith" and while this is obviously up to debate, if you are referring to the empirical period, most would probably point to the colloquially termed "Five Good Emperors" of the Nerva–Antonine dynasty, spanning across what made up the 2nd Century AD. During this time we have some great leaders that all had one thing in common, they were selected rather than the direct heir's of their predecessors. Because of this, Rome was fortunate to have competent leadership rather than the flipping the genetic coin, and under such leadership, the empire flourished. The stark contrast is the last emperor of the dynasty, Commodus, who ranks amongst the worst emperors. Despite being the son of a very good emperor (yes sorry I know its subjective), he turned out to be a crazy person. But fortunately for Rome, despite the horrors of the various bad emperors (read Caligula, Commodus, etc), the empire had already reached hegemony in the Mediterranean at this point. Especially during the Crisis of the Third Century, there were a lot of points when bad leadership could have spelt the end of the empire, but Rome got through it. Despite threats on the frontier and periodic wars with Persia, Rome's dominance in the region, combined with some luck, allowed it to overcome the horrific leadership of its bad emperor's, while allowing it to flourish under its good ones. From Marius, to Alexander, and Diocletian, Rome was lucky to have some great leaders when they needed it and were lucky enough to get through the bad ones.

There is obviously no easy answer to your question, but hopefully that can help shed more light on it. I tried to source some specifics but my overarching source is the fantastic "The History of Rome" Podcast which is a great introduction to the Roman Empire. I have seen through this series and other readings that it was really the Roman ability to adapt that gave it such longevity. The shifts from the Monarchy, to Republic, to Principate, and to the Dominate all represent such vastly different political structures yet we consider it all Rome. I can't comment on how much all of these other groups tried to style themselves on Rome, but the linguistic legacies of the "Kaiser" and "Czar" (both translations of Caesar) all point back to the amazing longevity of Rome.