is it true that archers needed to be highly mobile and capable of quickly shooting while at a jog?

No, not true in the least. While pinpoint accuracy wasn't required for battlefield archery, shooting while jogging or running would be so inaccurate that it would be a waste of arrows. Archers, as lightly armored troops, might be required to run to another position after shooting a volley to two, but they did not move while shooting.

is it true that archers held arrows in their bow- or arrow-hand while shooting?

Yes, this happened, although I'm not sure why this video mentions it as some kind of legendary secret technique.

is it true that archers were required to fire at quick speeds, like the quoted 3 arrows in 1.5 seconds?

Maybe as a training exercise, but frankly I've never read anything that would indicate any particular speed of shooting was required. Commanders would definitely want skilled and experienced archers, but it was more important to shoot in coordinated volleys on command than to simply dump arrows as fast as possible. Remember, archers were not shooting as individuals, but as part of a unit with hundreds or even thousands of other archers.

Pretty much all of this video is goofy trick shots and ludicrous scenarios. No historical archer (with the possible exception of drunk teenagers) was catching arrows out of the air or jumping and spinning while drawing a bow. This video is the equivalent of all the Kendo teachers who claim that they're teaching ancient secrets passed from samurai for a thousand years or whatever. Just like those kendo teachers, it's bullshit intended to make the listener feel important and cool for being taught these supposedly "lost techniques." /u/Valkine did a great job of answering your questions about early firearms replacing bows in this thread, but feel free to ask if you have any more specific questions.

Many cited the introduction of plate armor, the pre-eminence of personal firearms and the decline in hobby archery as reasons. What does current historiography have to say about the causes of the decline of military archery?

Hello, this is one of the most commonly asked questions on this subreddit. So luckily for you you have all the answers at your fingertips waiting in our FAQ!

http://www.reddit.com/r/AskHistorians/wiki/war#wiki_early_firearms

Included in this topic would also be, directly below this section, "Transition from Bows to Firearms", directly above the subsection "Guns vs Bows/crossbows and longbows in the age of the gun"

These threads I point out in particular:

• If early guns were so inaccurate and took so long to load, why did they so quickly replace crossbows, archers, and other earlier forms of projectile weaponry?

• A friend made this statement to me a couple of days ago, "The English Longbow was a superior missile weapon to everything up until the American Civil War when repeating rifles started showing up." Was he right?

P.S.: It should be noted he's talking about Civil War calibers but smoothbore muskets actually used a heavier caliber than those rifled muskets. So early muskets would, basically, transfer more kinetic energy and have more muzzle velocity.

• The bow is better than the musket - why did Napoleon not use archers?

• Why were primitive firearms used when bows and crossbows were better in every way?

It's not one thing but a conglomerate of things like you mention, in tl;dr. Armor made bows and other projectile weapons less useful. Guns are easier to make and easier to maintain and easier to train with. Guns serve a shock factor that bows can not replicate. By the time of the ringed and plug bayonets they also served a function no other projectile weapon could -- they could double as a pike.