WRT your questions about speed: doctrine created in 1856 by an artillery review panel, consisting of William French, William Barry, and Henry Hunt favored rigidly controlled rates of fire to approximately 1 round per minute. Hunt is reported to have considered firing faster than that to be a sign of cowardice, as the artillery crews would exhaust their supplies more quickly and have to retire to re arm.

I would speculate that the slower rate of fire would also allow better accuracy, as the smoke would have more time to be able to drift from the cannon and results of the shot easier to spot. At Gettysburg, Confederate artillery firing rapidly during Pickett's Charge overshot the Union lines and were not able to correct their shooting.

Asking whether a nation's military force was the "best in the world" invites supposition and opinion. I'll try to stay away from both.

Coming out of the Mexican-American War, US military leaders credited US artillery forces, especially light artillery, with being decisive to several major battles of that conflict. In most cases, American light artillery was able to move quickly around the battlefield in support of the infantry. As the Civil War approached, however, the introduction of rifling to both infantry and artillery weapons meant the use of artillery would be largely defensive. The increased range of infantry rifles would prevent artillery from being used in close offensive engagements.

We might conclude Union artillery gained its fearsome reputation in no small part due the offensive tactics favored by the Confederacy. Massed artillery firing at exposed charging troops was ferociously effective. Pickett’s Charge is a perfect example of this: Union artillery started firing at the Confederate charge at 200 yards, firing their last munitions when the charge was just 20 yards away from the guns. Pickett would lose half his brigade to enemy fire. Confederate forces also suffered significant losses to Union artillery at battles like Shiloh, Malvern HIll, and Chancellorsville. When massed in defensive positions to fire at exposed troops, however, Confederate artillery was equally devastating at battles like Fredericksburg and Chancellorsville.

When called upon to support offensive actions, Confederate artillery often came up short. At Gettysburg, for example, the Confederates used 142 guns to support Pickett’s Charge, facing 103 guns on the Union side. The plan called for Confederate howitzers to move up with the infantry, but they could not be found in time for the charge. Later, Longstreet commented the Confederate artillery barrage was ineffective, and a Union artillery officer at Gettysburg noted of the Confederate artillery fire “beyond the noise which was made no great harm was done.”

It’s clear Union artillery was superior to Confederate artillery in terms of equipment, tactics, and overall effectiveness, but how would Union artillery compare to European artillery forces? I’m not deeply familiar with this area, but I would encourage you to study the British and Prussian artillery forces. European powers of the mid-19th century, led largely by the Prussians, were modernizing and professionalizing their armies, institutionalizing training, planning, and military education.

Dupuy notes that at the outset of the Austro-Prussian War of 1866, Moltke the Elder (Prussian Chief of Staff) thought Austrian artillery was more accurate and longer-range than Prussian artillery, but he believed that Prussian training would “offset the slight Austrian technological superiority in cannon.” Most Austrian artillery guns were muzzle-loaded pieces, while about two-thirds of Prussian guns were modern breech loaders.

While I could find no information that Prussian artillery outperformed Austrian artillery (I’ll leave this to those with more detailed knowledge), we do know that by the time of the Franco-Prussian War in 1879, Prussian artillery out-classed the long-vaunted French artillery very easily. Dupuy quotes French military historian and lieutenant colonel Leonce Rousset on the state of Prussian artillery: “While our adversaries had for some time been equipped with steel, rapid-fire artillery pieces, we retained bronze, muzzle-loading cannon which had been outmoded even in 1859.” Rousset goes on in detail about the Prussians’ abilities to coordinate and mass fires while French artillery continually went into action in haphazard fashion. The Prussians had complete artillery superiority in that conflict.

Sources:

Attack and Die: Civil War Military Tactics and the Southern Military Heritage by Whiney and Jamieson, (1982).

A Genius for War: The German Army and General Staff 1807-1945 by Trevor N. Dupuy, (1984)

cases of artillery crews performing "trick shots"

From "A Sketch Of the War Record Of The Edisto Rifles" by William Izlar, 25th South Carolina Volunteer Infantry, about the Union siege of Fort Sumter in 1864:

"The Yankee artillerists at this time had become very expert, and I have seen them make some remarkable shots. A young signal officer by the name of Huger, who was stationed in Fort Sumter, one day went up to the top of the fort, when no firing was going on, and rested his telescope on the parapet to make observations. His whole body was hidden, only his head showing above the fort. The Yankees trained a gun on him from Cumming's Point and shot his head off. They frequently would cut the halyard of the flag as it was being lowered at sundown by a shell from one of their batteries. Such occurrences were too often to be mere accident."