Do strikes with weapons etched with runes inherit traits from those runes?

Question

The Conduct Energy free action states as a requirement:

Your last action or spell this turn had the acid, cold, electricity, fire, or sonic trait.

The Conducting rune grants a weapon etched with it the Resonant trait, granting access to Conduct Energy.

If I etched that same weapon with, say, a Corrosive rune, would it grant any Strikes made with that weapon the Acid trait?

If so, does that mean that the following sequences of events is legal, RAW?

• First action: Strike (with the acid trait)
• Free action: Conduct Energy
• Second action: Strike (with the Conducting rune now active)

If so, does this mean that the second Strike would deal x[W] + 1d6 acid damage + 1d8 acid damage, where x is the number of weapon damage dice and W the weapon damage die?

tl;dr Does having a Rune with e.g. the Cold trait on a weapon make Strikes with that weapon also inherit said trait?

My gut says yes, as this seems to be the mechanism by which magic weapons grant Strikes the Magical trait:

Ghosts and other incorporeal creatures have a high resistance to physical attacks that aren't magical (attacks that lack the magical trait).

Answer 1

No, with the latest errata and clarifications from paizo available here. In fact, this specific interaction with Conduct Energy is called out in the section for the Lost Omens Ancestry Guide that the ability comes from:

What actions qualify for the requirements of the resonant weapon trait's Conduct Energy action?

You can only use Conduct Energy with actions that have one of the required energy traits. Using an action other than a spell that causes damage with the listed trait does not necessarily qualify unless the action also has the trait.

For example, if you used the ifrit's Scorching Disarm action, you could channel fire energy into your weapon via Conduct Energy, as Scorching Disarm itself has the fire trait.

However, if you made a Strike with a flaming weapon, the Strike action does not have the fire trait, so you couldn't use Conduct Energy.

So Strikes made with rune-etched weapons don't inherit the associated elemental trait. Such Strikes are still considered magical, by this line under the Damage Types section:

Furthermore, most incorporeal creatures have additional, though lower, resistance to magical physical damage (such as damage dealt from a mace with the magic trait) and most other damage types.

Interestingly, the original printing of the Core Rulebook had a contrary section to this on page 451, Damage Types and Traits, quoted below. The section was removed silently in the second printing.

Damage Types and Traits When an attack deals a type of damage, the attack action gains that trait. For example, the Strikes and attack actions you use wielding a sword when its flaming rune is active gain the fire trait, since the rune gives the weapon the ability to deal fire damage.

Finally, you wouldn't deal quite that much damage in any case. As @WeirdFrog mentioned, conducting weapons only deal an additional 1d8, not xd8 as you've suggested.

Answer 2

Magical Weapons

To clear up a misconception first, a magical weapon is defined as any weapon with any rune on it. By necessity, the first rune must be +1 (or striking, technically, since it doesn't "require" a Potency rune like Property runes do... although you should probably get the +1). This can be seen on the Magic Weapon (equipment) entry.

Regardless, Rune traits probably get inherited

Nothing about Runes, including Fundamental Runes and Weapon Property Runes says that the Traits are part Strikes using that weapon (but nothing says they aren't, either). So why "probably"? Looking at Specific Magic Weapons such as Flame Tongue or Frost Brand, we can see that they do have the appropriate traits. As does the less-obviously-energy-attuned Blade of the Black Sovereign.