How did Hannibal manage to keep his soldiers in Italy for about a decade, away from their families? Did they go on a rotation back to Carthage ? Did he pay them?

There were no significant direct supply lines between Hannibal and Carthage (almost none of his soldiers were from Carthage, eventually many were even Italian) at any point during the Italian phase of his campaign, so his soldiers could not receive mail nor go home on leave, if such a logistical and organisational system existed in this period of warfare, which it did not. This lack of sea-route into Italy was a main reason Hannibal had to cross the Alps in the first place, and the only reinforcements he ever got from his brother Hasdrubal, that were destroyed at Metaurus in 207 BC, were also forced to use this route. Hannibal made repeated attempts to secure some control over a part of the Italian coast-line, the three unsuccessful battles he fought against Marcellus at Nola probably being related to this. He controlled Tarentum, which was on the SE coast of Italy for a few years, but he was never able to personally capitalise much on this as he was always busy campaigning elsewhere and the Romans took it back in 209. The Romans also managed to maintain control of the city's citadel with their remaining garrison for the entire 3 year duration the city was under Carthaginian control. Aside from this, which probably made it impossible to use the city as a strategic base of any kind, the fact that the few years Hannibal's subordinates held Tarentum saw no reinforcements from Carthage may also suggest that Rome controlled the ocean and Carthage's resources were too stretched, constantly paying and raising huge numbers of troops for the enormous Spanish campaigns.

Hannibal did his best to pay his men, but resources were tight in Italy. He was careful to offer financial incentives to motivate men before battle (Livy 30.33), but sometimes he lacked the money to pay them (30.12). He wasn't the only one with this problem, Scipio also suffering a near-mutiny over the issue of pay (30.24-9), and he needed to cough up the money to settle the whole thing down. Hannibal would have taxed the Italian population centres he controlled to the eyeballs to pay his troops, but no doubt their pay was in significant arrears by the end of the campaign. The fact that we hear nothing of any real mutiny against Hannibal is testament to how well-loved he was as a general.

Hannibal had no choice but to keep his soldiers away in Italy so long if he wanted to win the war and guarantee his own reputation, and if they wanted to be a part of his conquest, which would have been the most glorious in human history if it had succeeded, they had to stay with him. Hannibal's soldiers loved him, and despite desertion problems among Italians, soldiers from Spain and Gaul which formed a large proportion of his troops didn't necessarily have anywhere to go even if they did desert. Of course it was stressful for soldiers to be away from home so long, we can see in the mutiny suffered by Alexander the Great in the east that there was a point where even the best army got sick of it. But Hannibal's men (which through huge casualties and turnover would have been mostly a different group of people at Zama than they were at Cannae) stayed mostly loyal, and that's a positive reflection on their own determination but also Hannibal's reputation and performance. Of course, later in the war he used more and more Italian soldiers, prominently Bruttians, who ended up having to surrender back to Rome again, but many Italians may have even followed Hannibal over to Africa! To some extent, they may have had little choice after their 'crimes' against Rome.

Livy gave us this emotional piece of literature in the Late Republic, which has shaped the views of many regarding the support/lack of support of Hannibal by Carthage. This takes place shortly before Zama at Hannibal's recall from Italy:

"It is certain, that Hannibal made no farther efforts in Italy, for deputies came to him from Carthage, recalling him to Africa, nearly at the same time with Mago. Hannibal is said to have been thrown into the most violent agitation, and scarcely to have refrained from shedding tears, on hearing the words of the deputies. When they had delivered the orders, which they had in charge, he said,—“Now, indeed, they recall me, not in ambiguous terms, but openly, who have, for a long time past, been dragging me home, by refusing me supplies both of men and money. It is not the Roman people, so often discomfited and routed, that has conquered Hannibal, but the Carthaginian senate, through the malicious suggestions of envy; nor will Scipio exult, and pride himself, so much in this my disgraceful retreat, as will Hanno; who, unable to do it by any other means, has crushed our family under the ruins of Carthage.” (Livy 30.19-20).

Livy wrote this nearly two centuries after the fact, and it's probably just made up. The point he is making is that despite the skill and virtus of Hannibal, Carthage lacked the military and moral qualities of Rome and failed to support its commander, whereas Rome would have done so and was therefore the more worthy city. It's a moralising piece, but it does not take into account that Carthage either did not have a sea-lane to assist Hannibal with, and their resources were quite stretched anyway. There was probably not much more that Carthage, or Hannibal, could have done. The final great effort in Italy that might have changed things was Hasdrubal's crossing into Italy in 207 BC, but the two Roman consuls managed to give Hannibal the slip at concentrate on Hasdrubal to defeat him at Metaurus. Let's not forget that was Hannibal's error if it was anybody's, and probably his last real chance to change the tide of the war in Italy. The strategic and logistical limitations of Hannibal's war in Italy made the task of defeating Rome almost impossible, but he also made mistakes, and Carthage cannot be blamed for the all-but-inevitable result of Hannibal's defeat by the overwhelming resources and determination of Rome.