Yes and no. Ancient battles weren't the scrimmage portrayed by Hollywood, where every man breaks off into a separate duel. That's kinda counterproductive to unit cohesion, which was a centrepiece of the Legions, first off. Centurions were essential to that cohesion, hence why armies in Caesar's time were divided into unit blocks (centuries). Centurions were expected not only to lead from the front, but to take their own initiative when the battle called for it. This could prove to be problematic - such as at Dyrrhachium, when a few cohorts and Caesar's cavalry simply got lost:

After a stiff fight the fort was stormed, the legionaries hacking apart the barrier of stakes that blocked its main gateway. However, things then started to go badly wrong. Although Caesar's men were past the outer wall of the fort, there was another smaller enclosure within this and the garrison managed to cling on within this protection. Meanwhile, the cohorts of the right wing were unfamiliar with this stretch of the line and got lost, following a rampart that led away from their objective when they mistook this for one of the walls of the fort. Although puzzled that they had not encountered a gateway, the units kept going, and were followed by Caesar's cavalry.

By this time Pompey had responded, launching an immediate counter-attack with the five legions working to fortify his new camp, their approach inspiring the survivors of the garrison to renewed enthusiasm. A large body of Pompeian cavalry also headed towards Caesar's right wing, and the Caesarean horsemen dissolved into panic, fearing that their line of retreat back to their own lines would be cut off. The situation was confused, the panic quickly infectious. The right wing crumbled first, but as men saw this happening the rest of the attacking force also began to flee. Some men were trapped in the ditches around the camp as the cohorts dissolved into a mob and each man tried to force his way past his comrades.

...or it could be decisive, such as at Cynoscephalae: (quoting from Matyszak's Roman Conquests: Macedonia and Greece

The different fortunes of the two wings produced the inevitable result. A gap opened up between the Macedonians under Philip pressing on downhill, and the already tardy forces of Nicanor, struggling to hold the Romans on broken ground. The connection between the two phalanxes had literally become unhinged. This was noted by a nameless but quick-thinking military tribune on the Roman side. Every legion had about half a dozen of these tribunes, often young aristocrats in their first military command, or experienced soldiers with previous experience in commanding auxiliary soldiers. This commander suddenly noticed that when looking left he was looking at the backs and shoulders of Philip's men as they advanced downhill.

In battle, military tribunes were expected to use their initiative, as this man decisively did. He pulled together a scratch force of some 2,000 men from the reserves and the right-hand portion of the forces fighting Nicanor's phalanx and took them in a charge across the centre of the battlefield to fall on the left flank and rear of Philip's phalanx.

This attack was a cruel thing to do to a tight-packed body of men intent on moving forward. The last thing a phalangite with a 16 foot pole wedged between the ranks of his fellow soldiers wanted to feel was the presence of a much more heavily armed legionary at his back. And sadly, due to the unplanned nature of the battle, Philip had been unable to put in place the peltasts who usually had the job of guarding a phalanx from such assaults.

The battle there ended in a decisive victory for the Romans, as you can imagine.

So how much control did the general in charge actually have when the die had been thrown and the battle had been started? Well, a decent bit, if he was a good general. The basic battle plan, and those who were engaged, were generally left to the intuition of the men in charge locally. The general, however, would not generally commit all of his troops at one time. He, especially in the Roman world, would generally be in a place where he had a full view of the battle. Reserves were held back to be deployed where they were needed most at a certain time. In the Republic, these reserves were more formalized - the entire army was divided into three sections, and only one would be engaged at the start of a battle - but even after the Marian and Augustan reforms, holding troops in reserve was a crucial talent for any general to have. If a wing began to crumble, it had to be quickly reinforced by fresh men, at the risk of losing an entire army.

A second, equally important role of a general was when to know when to commit himself. The presence of a general among his men was an enormously emboldening thing, and Caesar himself made liberal use of it. Having a general within their ranks often gave men a second wind, as it were, letting them shake off their losses more easily, or shaming them into compliance. One of the more famous occurrences is Caesar's performance at Alesia (in his own words - Gallic War 7.86,88):

Caesar, on observing these movements, sends Labienus with six cohorts to relieve his distressed soldiers: he orders him, if he should be unable to withstand them, to draw off the cohorts and make a sally; but not to do this except through necessity. He himself goes to the rest, and exhorts them not to succumb to the toil; he shows them that the fruits of all former engagements depend on that day and hour. [...]

[...] Labienus, when neither the ramparts or ditches could check the onset of the enemy, informs Caesar by messengers of what he intended to do. Caesar hastens to share in the action. His arrival being known from the color of his robe, and the troops of cavalry, and the cohorts which he had ordered to follow him being seen, as these low and sloping grounds were plainly visible from the eminences, the enemy join battle.

Another example was in Spain, after he had defeated Pompey, and he had to singlehandedly rally his wavering army (Appian, 2.104):

When battle was joined fear seized upon Caesar's army and hesitation was joined to fear. Caesar, lifting his hands toward heaven, implored all the gods that his many glorious deeds be not stained by this single disaster. He ran up and encouraged his soldiers. He took his helmet off his head and shamed them to their faces and exhorted them. As they abated nothing of their fear he seized a shield from a soldier and said to the officers around him, "This shall be the end of my life and of your military service." Then he sprang forward in advance of his line of battle toward the enemy so far that he was only ten feet distant from them. Some 200 missiles were aimed at him, some of which he evaded while others were caught on his shield. Then each of the tribunes ran toward him and took position by his side, and the whole army rushed forward and fought the entire day, advancing and retreating by turns until, toward evening, Caesar with difficulty won the victory. It was reported that he said that he had often fought for victory, but that this time he had fought even for existence.

Hope that answers your question about the efficacy of generals on the field! Plans changed, as you noted, and the generals often had to supply the solution to those problems themselves - because they were the ones who would take the credit for victory or the blame for loss.

This is a really hard question to answer, because as far as ancients go we don't really know. There are almost no first hand accounts from either soldiers or low level commanders such as the centurions of the Roman Legions.

What I can say is that in everything I've read about all battle from Alexander to Napoleon the gifted commander is the gifted campaigner. By that I mean the good commander has taken pains to train his army and his officers prior to the campaign, and then uses the movement of his army relative to the opposing force and the fixed strategic points (towns, passes, rivers and fords, etc.) to force battle on his terms and with his army best positioned to bring its most effective attack to bear. If he can choose the timing, location and disposition of both his force and the enemy's then historically speaking that commander has a much better chance of victory than the commander who has this dictated by his opponent.

Once battle was joined the commander would have been able to effectively control only the timing and location of the commitment of his reserves, and must have had to rely heavily on his officers and the training of his men to fight in their formations. Beyond that I think it unlikely that these officers were encouraged to use much initiative: Discipline and holding the line was much more important. In ancient armies, formations such as the phalanx and the century were famous for their discipline and cohesion which made them effective without the need of micromanagement by the commander.

Sorry for the non answer, but if you're interested I recommend reading up on the art of the campaign. It will help illustrate how many great commanders managed to win the battle before it was even joined. Sun Tzu's The Art of War is a good start point, and after that any good study of the campaigns of Napoleon, Alexander, Marlborough or even Lee and the other corps commanders of the Army of Northern Virginia in the early years of the Civil War. Subodai is a great one as well, though the Mongol art of war was so unique that it's not really comparable to any of the other great commanders' campaigns. I wouldn't recommend the Romans as much: It's hard to divest the facts from the propaganda from primary sources, and in any event the Roman military machine of the Late Republic and Early Principate was so strikingly superior to its opponents that even a mediocre commander stood a good chance of victory often enough.