You've touched on a few of the answers already, but there's a bit more to it in my opinion.

The early history is of course a bit sketchy, but I think it shows one of the Romans' core traits that would let them succeed the way they did: adaptation. After rolling over a tribe known as the Latins they adopted their language. When they got sick of their kings they adopted the idea of a republic. Many of their military ideas were adapted from people they encountered and fought. For example the design of the 'gladius', that typical Roman sword that you see, came from people they ran into when they were off conquering Spain. Later on when they fought Carthage, the supreme maritime power in the Mediterranean, they started with basically no navy at all. Apparently they found a Carthaginian ship washed up on the beach somewhere and reverse engineered it, mass produced it, and set the men to training on inland waters. Of course they knew they didn't have the experienced sailors to properly steer and ram these triremes into enemy ships as the naval doctrine of the day demanded, which is where their second trait comes in.

Looking at the Colosseum or any of the various ancient architecture in Rome today is a testament to the importance Romans placed on engineering. It was no different with the new triremes they created, which had the addition of a huge vertical boarding plank with a hook in it. The rookie Roman crews only had to get close the Carthaginians and release the plank which would slam down and hold the enemy ship, which allowed the Roman infantry to board and do what they did best. Of course there was also all of the glorious architecture, civil works like aqueducts, bridges, roads, and huge walls to keep the barbarians out. This infrastructure in turn supported their military conquests, making it easier for troops to get to where they were needed. The army itself used its men as both soldiers and builders. A large roman army encamped resembled a huge wooden fort ringed by hundreds of metres of palisades and other fortifications on all sides. A famous battle Julius Caesar fought in had half his army besieging a city while the other half on his outer fortifications protected them from a barbarian relief army. Another interesting fact to note is that the Romans were the first to create concrete, an art lost for over a thousand years after their demise, though I'm starting to digress a bit. Onto the next point.

Probably the most inconspicuous part of Roman success is their culture, or at least the part that refuses to give up when all other Italian or Greek peoples would. Ever heard of the term 'Pyrrhic victory'? The general it is attributed to, a Greek by the name of Pyrrhus, lamented that even though he won battle after battle against the Romans, they refused to reach a settlement and eventually were victorious. Again and again, especially throughout the republican era, the Romans would take devastating losses to the effect that any other state would begin seeking peace terms. Hannibal's crushing victory at Cannae, where perhaps 100,000 Romans were encircled and butchered after Rome had already suffered a string of disastrous defeats, was met by defiance when he demanded terms. A victorious Roman army would raze Carthage to the ground and sow it's fields with salt 50 years later (edit: so the myth goes).

There are perhaps a few more things I should mention. An advantage Rome possessed during the republican era was the way it could draw troops from its populace, all with a stake in the ensuing battle. The nature of its political system, where the politicians were expected to fight and die along with the soldiers (Can you image that today?), meant leaders tended to have the interests of the state at heart (although glory and personal wealth became more of a motive later on). During its transition to an Empire it was blessed with a series of good Emperors that allowed it to consolidate its territorial gains somewhat, and the inherent strength of Rome would allow its Empire to continue for quite some time despite all of the damage done thereafter.

Hope that provides some insight, these are just some opinions of mine with some historical factoids pulled from the deep recesses of my memory. For manpower statistics and geographic factors you'll have to ask someone else :}

For a somewhat less triumphalist response, Rome was situated in the middle of a very important trade route between north and south Italy, which was likely a reason for it developing as a large urban center relatively early compared to the rest of Italy. It was also subject to influence from the Greeks to the south and Etruscans to the north, which may have contributed to its cosmopolitan qualities. It is also possible that the threat of raids from the nearby Apennines region provided the impetus towards state development.