Skills that state "% weapon damage" will deal the elemental damage from a weapon. Skills that state "% weapon damage as (element)" will not. So the weapon's fire damage would come through with Spectral Blade, but not with Magic Missile or Ray of Frost.
If the skill deals fire damage, either "naturally" or from a weapon effect, it should activate Conflagration.
Some sources here, here, here.
Companion damage scales based on attack speed. This was confirmed by Bashiok and tested by theorycrafters, who arrived at these numbers*:
APS Companion Damage
1.00 30%
1.34 40%
1.40 42%
1.43 43%
1.60 48%
1.84 55%
*Calculated by looking at the in-game tool-tip, which changes based on the APS of the character.
Note: The wolf, being a shorter duration (25 seconds), does twice the %.
That is, the companion does the % of your weapon damage based on your APS. If it follows this logic of being a channeled DoT (which is what most theorycrafters have come to agree), then based on Bashiok's post, the tick rate (how often it attacks) increases as your APS increases. As for speed it attacks, it would behave like any other DoT in this situation. It would attack once per cycle for its inherent percentage, so if your APS is 1, it attacks once per second for 30% weapon damage. If your APS is 1.6, your cycle time decreases to 0.625 seconds per attack (1/1.6), so it would attack once every 0.625 seconds for 48% weapon damage. If it follows this model, your weapon damage would have nothing to do with companion damage, but its speed (and therefore indirectly, its dps) would affect companion damage.
As for the boar companion, it says on the wiki that it "strikes multiple enemies with every swing."
Edit: After some testing with a level 60 demon hunter, it appears that the percentages are roughly correct. For the ferrets, even though they are animated to attack separately, it seems the total damage is split between the two. The boar is really hard to judge, but as far as I can tell, the damage is the same as the others, just as a cleave attack that hits enemies in front of the boar for full damage to each.
All this information is gathered by personal experience or by experiences posted on Diablo forums that were confirmed by more than one person. It's difficult to get exact answers until everyone starts posting their findings, since Blizzard rarely explains how game mechanics work.
Best Answer
Bola Shot does Fire damage by default (unruned), you can add the Thunder Ball rune for lightning damage when the bola explodes (from the description, it still does fire damage on the initial hit)