I think jumping from a running horse it.s a little dangerous...i think they dismount after the charge. I don.t think that fighting from a horse is hard because you have the advantage of the height.

Remember, you're looking at around 800 years of development just in the west, and 1,800 years if you include the Roman Empire in the east. So the idea of 'Roman Cavalry' encompasses a range of types and tactics.

Roman cavalry during the Republic certainly fought dismounted, mounted, or as a mix of the two, and often integrated with light infantry and skirmishers. Polybius Histories, 3.65 describes the engagement on the Ticinus:

Publius put his sharp-shooters and Gallic horsemen in front, and bringing the others into line, advanced at a slow pace. Hannibal placed his cavalry that rode with bridles, and was most to be depended on, in his front, and led them straight against the enemy; having put the Numidian cavalry on either wing to take the enemy on the flanks. The two generals and the cavalry were in such hot haste to engage, that they closed with each other before the sharp-shooters had an opportunity of discharging their javelins at all. Before they could do so, they left their ground, and retreated to the rear of their own cavalry, making their way between the squadrons, terrified at the approaching charge, and afraid of being trampled to death by the horses which were galloping down upon them. The cavalry charged each other front to front, and for a long time maintained an equal contest; and a great many men dismounting on the actual field, there was a mixed fight of horse and foot. The Numidian horse, however, having outflanked the Romans, charged them on the rear: and so the sharp-shooters, who had fled from the cavalry charge at the beginning, were now trampled to death by the numbers and furious onslaught of the Numidians; while the front ranks originally engaged with the Carthaginians, after losing many of their men and inflicting a still greater loss on the enemy, finding themselves charged on the rear by the Numidians, broke into flight: most of them scattering in every direction, while some of them kept closely massed round the Consul.

Jeremiah McCall, in The Cavalry of the Roman Republic, makes the argument that the cavalryman was much more stable dismounted, that intermixing mounted and dismounted troops allowed more men to be engaged than a line of horses, and that the Romans preferred this relatively fixed, intermixed line to the freewheeling melee that would result from a purely mounted battle. The opposing argument is that they dismounted only after the lines clashed and the rear line only dismounted to get into the fight where they couldn't fit otherwise. During the late Republic, the Romans phased out the cavalry levy from the equestrian class, and relied more on auxiliary cavalry. The Roman cavalry of the Principate consisted of the 120 cavalrymen assigned to each legion for communications and reconnaissance, while auxiliary (particularly Germanic) cavalry took up the front-line cavalry role (who would often dismount or fight in mixed mounted/light infantry formations).