So while there are various "sharpshooter" battalions in the OOB of both the Confederate and Union armies, the idea of detailing them specifically with the task of going out and effectively assassinating officers on the field would have been considered dishonorable to most. Not only because many of the men on the other side of the lines might happen to be someone you served with, but because specifically murdering another officer was seen as crass by most men - whether it be from Sandhurst or West Point. I think many tend to forget this was still an era where men's personal honor (and indeed even an abstract kind of chivalry) were seen as very important.
As for why officers still led on horseback, well, no matter what marvelous inventions we might have with "field telegraphs" or even air balloons, signal flags, and heliographs, most men and officers will only be able to discern what is going on directly in front of them. Add to that the very real fog of war with smoke and fire from rifles and cannons fired by tens of thousands of combatants, and it's a very chaotic situation. Your men need to be able to see you to know what you want them to do, and being on horseback was the best way for a commanding officer to not only coordinate with his brigade/division/corps but it also meant he could ride to the scene of trouble when he was needed.
Essentially it all comes down to communications. It would be impossible for a general to coordinate a few columns of men across, say, 22 miles of front unless he knew they were all supposed to meet at a specific place. If any column came into trouble they would need to take the time to track down their commander, send a messenger to find their army commander, and then the army commander would need time to react to what had happened to one column. It's why armies concentrated for battle, and you don't see many battles in the civil war that exceed more than three miles of frontage. Beyond a certain range, timely communications with moving masses of men are pretty difficult.