Hello! The answer to the questions in your title is... maybe? If you have the time, please take a look at an earlier post on the Carthaginian Senate's general ill-will toward Hannibal and their lack of commitment to his campaign in Italy. However, the government must have held some interest in overall strategy, as they did raise armies and divert reinforcements to other theaters. Carthaginian senators also took part in the negotiations between Hannibal and Philip V of Macedon. There's an anecdotal story that, between Hannibal's return to Africa in 203 and the Battle of Zama, the authorities kept him and his soldiers busy by having them plant olive groves throughout the countryside; I suppose you could interpret this as an attempt to discourage a coup.

Was the government effective in military matters? I would say no. Like most representative bodies, the Carthaginian Senate was not a monolithic entity, and it contained various factions centered around prominent political dynasties and interests. On the other hand, I reckon that the vast majority of senators were chiefly interested in self-preservation and/or maintaining the status quo. What's my evidence? According to Appian (Hisp. 4), sometime after the First Punic War or the Libyan War, Hamilcar Barca (Hannibal's father) was brought to trial by his enemies; in fact, the Carthaginian government regularly scapegoated returning generals through the Court of One Hundred and Four. In Hamilcar's case, the prosecution ultimately failed because other "leading men" stepped in on his behalf. Another source, Diodorus (25.8), states that Hamilcar was popular with the common citizenry ("the lowest sort of men") before later turning to demagoguery. My condensed logic follows as such: in the aftermaths of the First Punic War and the Libyan War, two calamities that revealed the many shortcomings of Carthaginian political leadership, some senators would have found it politically expedient to support Hamilcar the war hero; this was no longer the case midway through the Second Punic War, when Hamilcar was already dead and Hannibal's operations were getting nowhere.

That's mostly speculation, of course. It will suffice to say, according to our sources, Hannibal did not enjoy much support from the political leadership in Carthage, and this limited how much help he received from home. For the Senate, politics evidently trumped the war effort. :/

We have only limited sources from the time, all of them Roman. However, based on what evidence we do have this seems to be a very plausible explanation. Hannibal, his father, and his brother basically forced Carthage into the war. Meanwhile in Spain they had essentially set up their own personal Empire where coins and loyalty were all owed directly to their family. Now put yourself in the shoes of the Carthaginian senate. Here you have a hugely powerful, immensely wealthy, and endlessly aggressive member of your aristocracy. Carthage themselves had no idea the devastating impact the eventual peace would have on them, and if Hannibal succeeds too much what is to stop him from sailing back to Carthage at the head of a large army and demanding to be declare king? If he had succeeded in shattering Rome's power in Italy and fractured the peninsula back into it's 4th century political divisions, there would be no power in the western Mediterranean that could stand up to him, Carthage included.

The actions undertaken directly by Carthage (and not Hannibal or his brother) during the war focused instead on attempts to retake what was lost in the First Punic War, which I think reveals the motivations of the Carthaginian leadership outside of those two brothers pretty well. Particularly important was Corsica and Sardinia, which had basically been stolen at the conclusion of the First War.