AFAIK you can equip any ability/trait to any class.
If the ability increases fire crystal drop rate it would be silly to limit it to blacksmith, you can't kill much stuff with rocks.
Nope, plenty of abilitys are class specific ex: Tranquility or discipline specific ex: Brand of Fire.
In regards to the specific ability the OP was asking about, I'm assuming you're refering to Brand of Fire. This ability can be used by any DoH (Armorer, Carpenter...) but not by DoW, DoM or DoL. Also, it doesn't do quite what you think. It doesn't increase rate of crystal/shard drops, it improves the stability of the fire element while crafting. You see, it's possible for an element to become 'unstable' while crafting, possibly resulting in an explosion (or something catastrophic, can't remember the exact term) causing a very large portion of duribility to be taken away. This ability helps to prevent that in the case of the fire element.
Hope that helps.