Please consider that there is difference between what an archer could do and what one would do in a battle.

From Robert Hardy's Longbow:

How fast did they shoot and how many arrows were available? It is perfectly possible to shoot, out of a bow of 70 lb (32 kg) or so, 15 reasonably aimed arrows in one minute. Ten is no great matter. If the archers of Crecy could manage their much heavier bows as well as we can cope with lesser ones, and I have no doubt at all they could, then we can confidently allow them to have shot off ten arrows a minute each during an enemy attack. Prince Louis Napoleon considered that 'a first rate English archer who, in a single minute, was unable to draw and discharge his bow 12 times with a range of 240 yards and who in these 12 shots once missed his man, was very lightly esteemed'.

Take into consideration that Louis Napoleon (whichever one it actually is) is not from the time when the English Longbow was at its height.

The rest of the book goes into greater detail about the usefulness of the bow albeit a bit romanticized.

A few more quotations on arrow speed:

He recounts what Maurice Thompson once wrote:

'The marching commands and orders are those of the United States infantry. There can be no military display finer than that of a well-drilled, uniformed and equipped archery company. And at such short range as is needed in times of riots in our cities, no company would be more dreadfully effective. A well-trained archer will discharge thirty arrows in a minute and every arrow is death.' That astonishing claim is from a trained archer who knew and experienced the need for fast shooting, so we should believe that what he claims was true, at least for him.

From personal experience, he describes archers he has encountered during his years of involvement in the English archery scene:

I know of young archers who can handle weights well over a hundred pounds, as well as those who have trained themselves to shoot, with reliable accuracy, twenty and more arrows in a minute.

On the issue of rate of shooting in a battle:

Of course the number of archers engaged at any moment depended on the breadth of the attack, and on how many bowmen could see their targets. Mathematics of this kind are beguiling and dangerous but, if for any one period of two minutes, 3,000 archers could keep up the rate of 10 arrows a minute, the resulting storm of shafts would be 60,000. I do not suggest that such a number of arrows was actually shot off in any two minutes at Crecy or anywhere else, but I do suggest the total number of arrows shot off during the hours the battle at Crecy lasted must have been somewhere in the region of half a million.

Now, I realize that my answer is very English-centric but it probably was quite similar for other cultures. The biomechanics are quite similar for all kinds of shooting (with the possible exception of Japanese Kyudo-style shooting). A soldier of one ethnicity would not have significantly different work capacities, so there is only so much a body can accomplish.

If you are interested in what a modern-day record holder of speed shooting looks like, take a look at Lajos Kassai. Here one clip where he was featured on History Channel. 12 arrows in 17 seconds.

Here the Guinness World record recording - same feat as before but riding

Lastly, your questions on the effectiveness of the bow on battle lines are much harder to answer. There doesn't seem to be much consensus at all. One way to approach the subject would be to look at an arrows penetrative ability against different types of armour. Alan Williams' The Knight and The Blast Furnace is the best academic study on this topic we have. The results are quite unambiguous in that arrows could be rendered ineffective by sufficient armouring. These kinds of discussions however overlook tactical advantages volleys of arrows might have had.