Yes, mass volleys of arrows absolutely existed, and the delivery of such volleys was the main function of large groups of archers on the battlefield (in European warfare, at least). Foot archers didn't (usually) pick their own individual targets; they shot at larger enemy formations. The greater strategic purpose of archery on the battlefield is to harass and weaken the enemy, not shoot them to death (although that sometimes did happen).

1. What do you mean by "too many archers?" Ancient and medieval armies could bring formations of archers in the hundreds or thousands. Commanders would try and fight on terrain that allowed them to effectively field and maneuver whatever men they had brought.

2. Besides ordering archers not to shoot, there's no way to prevent archers from injuring their own troops when they loose their volleys. When the enemy had moved close enough to engage hand-to-hand, archers might be sent to flank around and shoot into the sides or rear of the enemy army, where there were no friendly troops to be accidentally hit. In late medieval England, it was common for archers to simply charge forward and take part in the melee themselves. When they backed up or provided flanking support for lines of men-at-arms, it was an extremely effective tactic. All the famous English longbow victories were won not only by mass archer volleys, but hours of brutal hand-to-hand fighting.

3. I'm not quite sure what you mean by "a useful defense mechanism." Mass archery was a defense in the sense that (when used effectively) it could disrupt enemy advances or slow down a charge (in rare cases, even halting a charge entirely). Archery was an excellent defense mechanism if the archers were protected by terrain, field defenses, and/or heavy infantry. Assuming that the enemy could be lured into assaulting such a position, that is. The major issue there is that it requires an opponent who can be manipulated into launching a forward attack.

Not showers necessarily, but in Japanese warfare at least, arrow volleys were used. Similar to the reasons why muskets were fired in volleys, having archers fire like that gave them a greater chance to hit moving targets and do more damage to group formations.

If OP don't mind me piggybacking on this question, I'd like to know what historians think of this video, in which an archer was able to shoot at incredible speeds, and claims that his technique is based on historical battlefield archery.

Just to add to the answers here, Matthew Barnett writes in "Fighting techniques of the medieval world" that an archer could loose a third arrow before the first had struck its target.

According to The Face of Battle (John Keegan, 1976), at Agincourt:

"...the archers, both those in the large groups on the wing and the two smaller groups in the central wedges, the chance to prolong their volleying into the French ranks. The range was progressively shortened by the advance, and the arrows, coming in on a flat trajectory in sheets of 5,000 at ten second intervals, must have begun to cause casualties among the French foot."

So I would say that showers of arrows really could exist.

Keegan goes on:

"... some of the arrows must have found weak spots in the visor and at the shoulders and, as the range dropped right down, might even have penetrated armour itself. The 'bodkin-point' was designed to do so, and its terminal velocity, sufficient to drive it through an inch of oak from a short distance, could also, at the right angle of impact, make a hole in sheet steel."

More, about the opening of the battle:

"Each [archer] disposed his arrows as convenient. He would have a sheaf, perhaps two, of twenty-four arrows and probably stuck them point down into the ground by his feet. The men in the two front ranks would have a clear view of the enemy, those behind only sporadic glimpses: there must therefore have been some sort of ranging order passed by word of mouth."

"... when the shout went up or the banner down, four clouds of arrows would have streaked out of the English line to reach a height of 100 feet before turning in flight to plunge at a steeper angle on and among the French men-at-arms opposite. These arrows, cannot, however, given their terminal velocity and angle of impact, have done a good deal of harm... But one should not dismiss the moral effect of the arrow strike... the sound of their impact must have been extraordinary cacophonous, a weird clanking and banging on the bowed heads and backs of the French men-at-arms... If any of the horses... were hit, they were likely to have been hurt, however, even at this extreme range, for they were armoured only on their faces and chests, and the... arrow would have penetrated the padded cloth hangings which covered the rest of their bodies."

The Face of Battle, for those who haven't read it, is a fascinating discussion of what the experience of being in the front lines was like at Agincourt,Waterloo and the Somme.

Herodotus does make mention of this action in his history of the Peloponnesian war. A Thrachinian makes the comment “Such was the number of the barbarians, that when they shot forth their arrows the sun would be darkened by their multitude”. What is particularly interesting about this extract is that Dieneces does not laugh this statement off, nor does he discredit it in any way. He responds with, "Our Thrachinian Friend brings us excellent tidings. If the Medes darken the sun, we shall have our fight in the shade".

Source: Herodotus, The History, Chapter 226.

Available here

http://www.documentacatholicaomnia.eu/03d/-484_-430,_Herodotus,_The_History,_EN.pdf