I recently have listened to a podcast (The History of Rome) and a book (Ghosts of Cannae) which have dealt with the Second Punic War. Of course both of them touched on Hannibal's use of war elephants, but they gave very different impressions. The History of Rome suggested that the elephants were very effective in battle and used by Hannibal to destroy or disrupt large parts of the Roman army. Ghosts of Cannae seemed to indicate that the elephants would almost immediately get scared during battle and start trampling Carthaginians and Romans alike making them tactically useless. Both sources seemed to agree that the elephants served a propaganda purpose as the Roman survivors would report facing an army of the biggest animals they ever had seen.
One one hand the Ghost of Cannae gives a more sober and less fanciful assessment of the situation that I am biased to agree with. On the other hand, its hard for me to imagine a man as skilled as Hannibal would not recognize their tactical uselessness. What is the consensus of how effective these war elephants really were?
First things first: Hannibal's war elephants were of a species (or subspecies; I'm not a biologist) of African bush elephant which is now extinct. This species was smaller than the African and Indian elephants we know these days. Moreover, Romans had already encountered elephants when fighting other hellenistic armies. Pyrrhus (of 'Pyrrhic victory' fame) had reportedly used elephants to great effect against Roman cavalry. While that first Roman encounter with war elephants didn't go particularly well, that was over 60 (!) years before Hannibal invaded Italy. So prior experience with encountering elephants reduced their shock effect on the Romans.
According to Polybius, Hannibal crossed the Rhone river in France with 37 elephants. It is with these elephants that Hannibal entered the Alps. There is a brief reference in Polybius (III.52) that "barbarians" (Gauls) in the Alps didn't dare attack the parts of Hannibal's column where the elephants were because they were "terrified by the strangeness of their appearance". The crossing of the Alps was disastrous for the health of the elephants though, and it seems they were weakened quite a bit. Polybius reports that elephants did fight at the Battle of the Trebia river (III.72) but he doesn't specify how many. Hannibal divided the elephants equally over his army's wings, to protect the flanks. The elephants and cavalry on the wings defeated the Roman wings in the battle, but Polybius does not mention any panic or such caused by the elephants.
After the battle, all but one of the elephants died due to a combination of their weakened state from the mountain crossing and cold weather following the battle. That last elephant carried Hannibal through the marshes on the way to Lake Trasimene (III.97), but it too perished some time before the Battle of Cannae. In 215 B.C. Hannibal received his only reinforcements of the war, which included 40 elephants. I have no information on how they were used though (which also means I have heard no reports that they were exceptionally effective or ineffective).
Hannibal's largest deployment of elephants occured near the end of the war at the Battle of Zama, in North Africa (near Carthage). Over 80 elephants were placed in front of his whole force. Scipio, the Roman general, had prepared for this though and had ordered his troops to make gaps in the lines. When Hannibal gave the order for the elephants to attack, some of the elephants panicked and attacked Hannibal's Numidian allies. The other elephants charged the Roman lines, but took the path of least resistance - charching through the gaps in the lines instead of attacking the Roman legionaries. (Polybius XV.11)
With this narrative in mind, let's get to answering your questions. The elephants were pretty effective in battle, though they were in no way an "I-win" button. They had the nasty potential of panicking and turning on anyone nearby, including their 'own' soldiers, but this evidently didn't happen so often as to negate their usefulness. The propaganda-effect on the Romans was limited. By the Second Punic War the Romans had been fighting hellenistic armies for a century or so, and had encountered elephants with some regularity. No soldier would be happy to encounter one in battle, but they knew what to expect and by 203 B.C. at least they were experienced and disciplined enough to even let them into gaps in their own lines.