All classes are associated with a specific weapon. However this does not mean other weapons cannot be used as long as the character meets the requirements, and there is currently no benefit for using their "native" weapon either.
Weapon damage is applied only during weapon attacks, but stat bonuses from weapons, like with all equipment, are always in effect.
A shield or dagger may be equipped in combination with a main 1-hand weapon. 2-handed weapons will occupy both left and right-hand weapon slot, preventing these options.