According to Bashiok:
...faster speed weapons consume more resource but may do more damage. Channeled skills might still be better with slower weapons though because on a channeled skill, you have as much mobility as you want, so the "mobility" and "overkill" advantages of faster weapons are eroded with channeled abilities, so all you’re left with is increased mana cost for increased damage.
So yes, a faster weapon will cause more skills to tick faster, resulting in you dealing more damage over time, but will result in you using up more resources as you are channeling faster.
Channeling mechanics
Let's use Ray of Frost as an example,
Cost: 20 Arcane Power
Effect: Channel a beam of frost that deals 215% weapon damage as Cold.
Since this is a channeled skill. The resource cost ticks by 10 and drains that much Arcane Power at twice the speed of your weapon. If your attack speed is 1 attack per second, instead of draining 20 Arcane Power once per second, it will drain 10 Arcane Power every half seconds.
Additionally, Since Ray of Frost is a channeled spell. It is broken down into many small attacks that deal a total of 215% weapon damage based on your attack speed. If your attack speed is 1 attack per second, it will deal a total of 215% weapon damage per second.
tl;dr: The faster the weapon attack speed, the faster the drain; slower the weapon attack speed, slower the drain.
Damage-dealing
Assuming two weapons have the same damage, the faster weapon will use up resources faster than the slower one when channeling, but the faster weapon will take less time to do full damage than the slower one.
An axe at 15 damage and 1.0 attack speed will drain slower. (15 DPS)
A wand at 15 damage and 1.5 attack speed will drain faster, but will do
the same damage faster. (22.5 DPS)
For attacks that hit multiple enemies, Yes.
I haven't explicitly tested all on-hit effects. I will update this answer as I do.
Life on hit: Confirmed. With a +2 life on hit weapon equipped, I often saw healing popups of +6, +8, +10, etc while using area attacks that hit multiple targets.
Chance on hit to fear: Confirmed. Using Deadly Reach I saw two enemies get feared and run away.
Best Answer
After some testing it does appear that multiple hits by using Magic Missile's 'Split' rune will trigger the effects of the Prodigy passive ability multiple times. With this in mind in conjunction with this thread relating to Electrocute, I believe it safe to state that the 'Prodigy' passive will proc every single time one of the signature abilities hits. This should mean that the Spectral Blades ability is overpowered for Arcane Power regeneration (at the cost of standing in melee range of course).
After some further testing with Spectral Blades I can confirm this is actually how the Prodigy passive works (one attack with Spectral Blades on a group of 7 enemies resulted in my Arcane Power going from 'nearly empty' to 'nearly full' in one attack).
My methodology for testing:
What I found:
Conclusion:
The Prodigy passive ability does exactly what it says it does. When you deal damage with a Signature ability with the Prodigy passive enabled, you will get 4 Arcane Power per enemy that is damaged during that cast.
For example; in the instance that you're using the Magic Missile ability with the 'Split' rune, you will gain 4 Arcane Power for missile, under the assumption the missile hits an enemy. If each of the 3 missiles hit the same enemy that still counts as three hits and you will gain 4 Arcane Power per hit accordingly.
This stacks with the equivalent runes on the Signature abilities themselves, so you can use Prodigy and the Siphoning Blades rune for Spectral Blades, for example.