The General's role largely depended on three things: 1. Their actual rank as general, 2. their role in the particular fight, and 3. their own personality and how they decided to enforce that role.

As to the first part, the higher the rank, the less was, generally (see what I did there?), the proximity to the actual fighting. This made practical sense, since the general in charge of his unit needs to be alive and capable of directing the battle, which they definitely did. Battles are always fluid, and the plan will virtually never work they way you plan it, no matter how carefully you do so. As circumstances change, units need to be shifted to fill gaps, moved to a flank, or withdrawn. Usually this is done from a Headquarters. As armies converged on a battlefield, usually a specific place would be selected that was advantageous, perhaps as a structure, like a house, that was close enough to all flanks to not be too out of the way, but offer a good location from which to direct the forces. Such as the Leister Farm house that served as General Meade's HQ at Gettysburg, or the McLean House at Manassas used by PGT Beauregard, or Cherry Mansion in Savannah, TN, where Grant was headquartered for Shiloh. If you see these, and other locations, on battlefield maps, they are typically close to the center of the lines, but behind them, so as to be able to direct from multiple angles. This is critical to the battle effort, as they only have the use of runners to convey messages. You have to recognize that many of these battlefields are huge in scale, with oftentimes pushing 75,000-100,000 people on one side. At Gettysburg, Meade's Army stretched about 1.5 miles, with interior lines, but Lee's were stretched over 3 miles, in a semicircle. This meant that for information to get passed, it had to go the long way around. This is also a large factor in why battles so often went off the rails almost as soon as they begin: its nearly impossible to coordinate tens of thousands of people across multiple miles of terrain to all be ready to go at exactly the right time without direct lines of communication. Added to this is the death of commanders, incorrect or misheard information, or multiple reports that contradict one another. The wrong decision, even for the right reasons could be disastrous. Take Chickamauga, for example. In the dense woods of northern Georgia, the Union reported a gap in the lines, so Rosecrans ordered the gap closed by shifting troops to fill it. Turns out, no such gap existed, but the moving of troops created one, which happened to be exactly where Longstreet's men broke through. The General in command needs to be close enough to be able to make decisions quickly, and get that information to subordinates, but even then, it may not be enough.

But being further behind the lines was in no way assumed to be less perilous a position. During the Battle of Chancellorsville, arguably what allowed the Confederates to defeat the Union was that Union General Hooker had been severely wounded by a cannonball. Generals, even in higher ranks, oftentimes suffered high casualty rates. If we look at major Confederate commanders in both eastern and western theater, Longstreet and Joe Johnston suffered major injuries as heads of either a complete army or a Corps of it, while Jackson, A.P. Hill, Leonidas Polk, and Sidney Johnston, to name a few, were killed. Even Lee had a close call at Fredericksburg when a cannon burst nearby. Union General John Sedgwick discovered the hard way that long distance shots can be as deadly as close ones, when he was killed from several hundred yards away at Spotsylvania.

Officers were expected to lead their men, and the larger the force involved, the higher the rank of the officers doing the leading. At Pickett's Charge on the third day of Gettysburg, Major General Pickett with his staff were watching the attack take place, but his subordinate brigade commanders, led most notably by Kemper, Armistead, and Garnett, actually went into the charge. The disastrous result was the loss of practically the entire division's effective strength, and the death or capture of three general officers. Franklin, TN was an worse defeat of the Confederates, losing 5 generals killed, including arguably their ablest commander, Major General Patrick Cleburne. Through the war, the Union lost around 50 generals (brevet or otherwise) to combat, the Confederates lost more than 70. But sometimes the decision of the general in question was to blame. At Shiloh, the commanding Confederate General, Albert Sydney Johnston, tried to rally his right flank by leading them personally, and was hit in the leg and bled to death on the field. While there's little doubt the Union would have won anyway, its probably likely that at the very least, if Johnston had survived, he would have been a better liked commander than the eventual Bragg, even if he perhaps would not have been more effective (speculation is of course all we have on that). Famously, or infamously, Stonewall Jackson scouted his forward lines at night outside of Chancellorsville, and was shot by friendly fire on his return.

There were, of course, some generals who got into the actual fighting itself more often, and these were almost always cavalry commanders. Nathan Bedford Forrest is certainly the most notable example of this, as he was typically one of the first men in any charge, and reportedly killed at least 30 men and had almost as many horses shot out from under him. JEB Stuart is another example, and was eventually killed at Yellow Tavern in 1864. Brevet General George A. Custer made a name for himself in the same way as a Union cavalry officer, so much so that following the war he was given a solid rank of Lt. Col, at a time when it was not uncommon to serve in the same officer rank for twenty years or more.

So ideally, the commanding generals needed to stay behind the lines, to coordinate effectively, but even this was not without its dangers. Brigade and Division commanders were many times called to be put into action themselves, and some cavalry officers did a variation of all three, stay back and direct, lead men into the fight, and do the actual fighting.