It's often been said that Rome's military success was in no small part due to its willingness to absorb huge losses and keep going, i.e.: Cannae, when any other ancient state would give up. Where did this never-give-up mentality come from, and what gave it the ability to raise such large armies relative to other ancient states?
Your first question is not truly answerable, other than to speculate on the effect Rome's early history had on them. We know they were subjugated by foreign kings, sacked by Gauls, and in near constant conflict with their neighbors from the very early days of the city's existence. However, none of these completely explain the ethos and stubbornness of the Roman character, something even their contemporaries did not really understand.
As for your second question, the answer is two-fold. First, we have to consider that Rome itself wasn't fighting Hannibal alone, but rather, had almost the whole Italian peninsula to draw upon. Roman armies included a large contingent of allies (theoretically for every Roman legion there was an accompanying allied legion) so while Cannae was a seemingly devastating loss for Rome alone to absorb, it was not such a devastating loss for the whole of Italy to absorb.
However, just as important is the reasons why they could raise such manpower even if we only consider the half loss. This is because Rome, as a military-minded Republic, had advantages most of their contemporaries lacked. One of those advantages was a large pool of at least moderately trained, but very possibly well trained and veteran warriors, men to draw on for their armies. Between 400BCE and 146 BCE Rome's armies were virtually on campaign non-stop every summer. Indeed, one of their most important democratic assemblies, the Centuriate Assembly, the one which elected most important offices and declared war, was based around the military organization of the citizens.
However, the fact that they were a Republic also gave them another advantage over most of their peers. You see, most other ancient states could only afford to lose one battle. Rome could afford to lose as many as it took, seemingly, to win a war. The reason behind this is pretty simple: if you are a tyrant or monarch, your hold on power is almost always going to be pretty tenuous. The same force that you use to defend your power against the Romans is the one you use to defend it against rebellion or local rivals. While the Romans could afford to lose entire armies and not have to worry about their whole government system coming down, Macedon or Pontus were not so lucky. This meant that her rivals were much more likely to sue for peace than Rome would, since a single defeat for Rome just meant another army needed to be raised. A single defeat for their enemies might mean the fall of a kingdom.