Elemental damage type of a weapon, so far as I understand it, is not applied to your attacks in any form. What you see as the abilities damage type is the type of damage it inflicts. Not the weapons damage type.
Yes, with one exception: the wizard passive Elemental Exposure. That's why you want your weapon's element type to be something different than the skill element you're using. Read the EE description here (last sentence): http://us.battle.net/d3/en/class/wizard/passive/elemental-exposure