What does the "Weapon Damage: X%" skill statistic actually mean?

Solution 1:

Yes, that means that 10% of your weapon's damage will be applied to this magical attack. Therefore, wielding a stronger weapon will yield more damage from this spell.

When it comes to dual wielding, the effects vary, there's a list of the Alchemist skills that benefit from dual wielded weapons on this Torchlight Wiki.

According to one Torchlight wiki, your weapon attributes impact your spell attributes in the following way:

• Mana and health steal are ''never'' triggered by these skills.
• "Ranged Damage" effects do not stack with any of these skills.
• Weapon speed has no effect.
• Knock back is hit-or-miss; works with some skills but not others.
• Fixed damage bonuses (i.e. "+100 ELEMENT Damage") are ''extremely'' buggy; on some skills they have 100% effect, on some skills they have none. Skills that supposedly add 20% weapon damage are in both categories, even when the damage bonus is on a weapon. They do not work on Pyre.
• Innate weapon damage is also rather hit-or-miss; several skills that say they do 20% of weapon damage actually do none.
• The damage bonus from weapons is not based on DPS; it uses only the actual listed per-hit damage values.
• Having said that, multiple damage types DO stack, so if you have a weapon with 100 Fire damage and 100 Poison damage, Ember Bolt gets 40 damage added to it, not 20. This is not how most people thought it worked.
• When dual wielding, only the highest damage appears to be used (this is per-hit damage, not DPS). They do not appear to be averaged, added, or alternated; the damage is consistent.
• But still, it's only 20%. You're almost certainly better off concentrating on getting a weapon with effects that stack with the skills, rather than the actual damage output of the weapon.
• Percent damage bonuses (i.e. "Increases ELEMENT Damage 10%") work on everything ... except Pyre.
• Critical Chance/Critical Damage bonuses work on everything ... except Pyre.

Unfortunately, actual data about the game mechanics and formulae is spotty at best and outdated at worst.