No, if you use Poison Weapon you probably don't need the passive Poison Damage (for melee attacks) or Poison Arrows. Poison Arrows do not require poison potions, just mana, but from what I have read, that spell is not as effective as Poison Weapon.
A poisoned arrow which hits the target can have the poison effect applied even if the arrow doesn't do physical damage. I tried clearing the orcs around Aleroth with a poisoned sword and did not notice any poison hits without physical damage (maybe the spin attack could poison with a miss, but I didn't go far enough south to hit large enough groups to be able to test that).
I don't think you can un-poison a weapon before the poison charges run out.
