Class specific weapons can provide some class only enchantments that you cannot get on standard weapons. Whether or not this is worth a trade off for damage, that is entirely up to you to decide.
For instance, you can find a demon hunter only 1 handed crossbow that provides hatred regeneration or a wand for mages that provides improved arcane pool.
Like James mentioned in his answer, the demon hunter is the only class that has a unique situation where you MUST have a ranged weapon equipped or you can't use his ranged attack skills. And his one handed crossbow is the only ranged weapon you can dual wield.
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)
Best Answer
I had the same issue when channeling Rapid Fire as a Demon Hunter. There seems to be two things in play here:
This curse is applicable at all levels, it's in game from Normal onwards. The simplest way I found to deal with it is either:
With respect to your last two questions about armor/resists, it's hard to tell at this point in time. There's very little information available about enemy abilities like this, and the lack of a combat log of any type makes it difficult to determine damage types. From my personal experience I don't think that using armor/resists to mitigate the damage coming from channeling is the way to go, just because of the sheer volume of damage that is done.
Blood Star related bug fixes in patch 1.0.3
Link to 1.0.3 Patch Notes.