The Battle of Ticinus was the first engagement of Hannibal's Italian campaign within Italy proper against Rome. I haven't read the books by Drake, but from what I can tell it may be very possible that he did combine the Battles of Ticinus and Trasimene as Ticinus happened next to a river rather then a lake, and Trasimene very clearly happened next to a lake. These water features both had a large impact on the battles, leading up to and after Ticinus both Rome and Carthage have to maneuver around the river repeatedly, at Trasimene the lake was a vital part of pinning the Romans in an ambush. So our sources very clearly agree about the nature of these geographical features. The author was simply mistaken, and without asking him its difficult to know why. However it is somewhat reasonable to suspect he confused it with Trasimene, as they both where important moments within Hannibals Italian campaign and Trasimene is more famous then Ticinus due in part to it just being a far more significant battle rather then primarily a reconnaissance action.

These battles can be difficult to understand without context, here is an answer I wrote a while ago briefly summarizing the first part of Hannibals Italian campaign for anyone curious.

Ticinus as I said was Hannibal's first action within Italy against Rome. After Publius Scipio found him on the river Rhone north of Massilia in Gaul, Scipio traveled back to Italy to warn Rome and prepare a defense against an invasion of Northern Italy. Hannibal once he had emerged from the Alps was desperate for supplies and safety for his army. Crossing the Alps had been incredibly difficult and his army has suffered a great deal both from the locals and the terrain. After emerging into the Po Valley he sent emissaries to the local Taurini, the Gaulic tribe who controlled the area. They did not respond with the alliance he had hoped for, Roman control was still fairly strong in the region and many tribes feared reprisal from the Romans. Hannibals response was to surround the Taurini capital and sack it. To be clear, I use the word Gauls but Gaul is used very broadly in this context and my usage is primarily because I find isolating the "Gaulic tribes" into their distinct ethnic groups confuses almost all lay men and they shared many similar cultural characteristics and the Roman's viewed them very similarly due to this, even if the personal stories of their people had many divergences from each other. Many of the Gaulic tribes Rome faced in Northern Italy where Celtic culturally and linguistically, and are referred to as such in certain sources. Enough of the boring history stuff lets talk about these battles.

To answer your second question,

2. For either battle, and just in Roman history in general, was there ever such an engagement, with the entire cavalry being dismounted (excluding times where the cavalry dismount before battle)

I will summarize what our sources tell us about the battles.

First lets again address the situation Hannibal has marched into entering Italy through the Alps. Our two primary sources for Ticinus are Polybius and Livy. They agree on most key parts of the battle, but differ on a few details like number of bridges on the Po. For now Polybius had this to say about Hannibals grand entrance into Italy.

After this, his forces having now picked up their strength, when the Taurini who live at the foot of the mountains quarreled with the Insubres and showed no confidence in the Carthaginians, he at first made overtures for their friendship and alliance, but on their rejecting these he encamped round their chief city and reduced it in three days. By massacring those who had been opposed to him he struck such terror into the neighbouring tribes of barbarians that they all came in at once and submitted to him. The remaining Celtic inhabitants of the plains were impatient to join the Carthaginians, as had been their original design, but as the Roman legions had advanced beyond most of them and cut them off, they kept quiet, some even being compelled to serve with the Romans. Hannibal, in view of this, decided not to delay, but to advance and try by some action to encourage those who wished to take part in his enterprise.

Hannibal's sack of the Taurini capital had been an important part of securing his position, the Gaulic tribes within reach of him no longer desired to oppose him and it gave him the freedom to secure his first victory against Rome. We also see in this passage that Polybius talks about Hannibal's intended Gaulic allies in Northern Italy, we see the stage get set with those same Gauls drawn into Rome's allied auxiliary. This was a very common practice through out Roman history, both in the Republic and during the Empire. As I mentioned in my other post, one of Hannibal's goals was to erode this allied support within Italy itself, as he believed that was the key to defeating the Roman Republic.

The cool thing about the wind up to Ticinus was how insane it was these two armies where fighting in Northern Italy at all. Neither Hannibal, who had just marched through an impossible mountain journey or Pubilus who had sailed most of the Southern French coast straight into Northern Italy believed the other was there. Neither commander initially thought the other was in Italy as the path through the Alps was treacherous and the path along the coast was very long by road or boat. They had just seen each other near Spain, at Massilia. So both Hannibal and Scipio incredulously confirmed each others existence in Northern Italy and rapidly moved to confront one another. Once Publius had confirmed Hannibal's location in Italy, he sent word to withdraw the other Consul Tiberius Sempronius Longus from the invasion planning in Sicily.

Before marching into each others orbit, both Consul Publius and Hannibal gave very dramatic speeches to inspire their armies according to Polybius and Livy, Ill include links here and here for reading if anyone is curious. I do not know enough to opine on the accuracy of these speeches, and I would rather not speculate but they are a fascinating view into how contemporaries viewed the conflict. Part of what is so cool about the Second Punic War is the legend of it, and how it reinforced Rome's view of itself. The entire story of the war reads like an overly complex HBO script with blood feuds, insane mountain crossings, elephants and giant armies. Rome more then anyone loved to feed into the legend of the Punic wars.

So once their men had rallied both armies set out to advance along the Po river, on the northern bank facing the Alps. Hannibal marched west, and Publius marched east. In order to discern the location of the other army on the river, both commanders had the idea to reconnoiter in force. Now I've been talking mostly about the Po, which is a very important river in Northern Italy. The Ticino is a tributary of the Po, and is the river the battle is named for. Here is a nice illustration from wikipedia to help us picture it.

Here is Livy's account of the battle

After the necessary propitiation had been made Scipio moved out with a force of cavalry and light-armed javelin men towards the enemy's camp to get a nearer view and to ascertain the number and nature of his force. He fell in with Hannibal who was also advancing with his cavalry to explore the neighbourhood. Neither body at first saw the other; the first indication of a hostile approach was given by the unusually dense cloud of dust which was raised by the tramp of so many men and horses. Each party halted and made ready for battle. Scipio placed the javelin men and the Gaulish cavalry in the front, the Roman horse and the heavy cavalry of the allies as reserves. Hannibal formed his centre with his regular cavalry, and posted the Numidians on the flanks. Scarcely had the battle shout been raised before the javelin men retired to the second line amongst the reserves. For some time the cavalry kept up an equal fight, but as the foot-soldiers became mixed up with the mounted men they made their horses unmanageable, many were thrown or else dismounted where they saw their comrades in difficulty, until the battle was mainly fought on foot. Then the Numidians on the flanks wheeled round and appeared on the rear of the Romans, creating dismay and panic amongst them. To make matters worse the consul was wounded and in danger; he was rescued by the intervention of his son who was just approaching manhood. This was the youth who afterwards won the glory of bringing this war to a close, and gained the soubriquet of Africanus for his splendid victory over Hannibal and the Carthaginians. The javelin men were the first to be attacked by the Numidians and they fled in disorder, the rest of the force, the cavalry, closed round the consul, shielding him as much by their persons as by their arms, and returned to camp in orderly retirement. Caelius assigns the honour of saving the consul to a Ligurian slave, but I would rather believe that it was his son; the majority of authors assert this and the tradition is generally accepted.

Continued in next comment