Do cold damage affix in weapon increase my lightning arrow skill damage?

If so, what's the condition and calculation?

Bumpidy.

Unless I'm understanding totally wrong you are asking if an increase in cold damage would affect lightning damage? No

It would increase any source of cold damage, including cold damage from weapon attacks.

I'm asking about weapons.
Sometimes they have an affix like "2-8 cold damage".
I'm using my Lightning Arrow spell. I wonder if it increases my Lightning Arrow damage.

Bump...

Since you're using a bow, I'd say Lightning Arrow IS affected by the amount of mods that add elemental damage. But I'm not entirely sure.

For everything else goes:

Check the DPS on any given spell, it does not increase when you equip a weapon that has "Adds x Cold/Lightning/Fire damage", it only affects the weapon itself and any attacks performed with it.

Increases in spell damage on a weapon show as "X % Increased Cold/Lightning/Fire Damage" or "X % Increased Spell Damage"

Attacks use your weapon damage as a modifier for the most part, including any elemental damage you do (the exceptions will mention it). Spells do not touch your weapon damage, even the elemental modifiers.

Well, right now I tested it by using a magic ring with "9-18 Fire damage bonus".

When I hover my Lightning Arrow spell in skillbar, it shows the DPS of that skill. With this ring equipped, I gain 20 more DPS.

So I believe any kind of elemental damage bonus increases my LA damage?

Items that have 'added x damage' will work with ATTACKS.

So in this case, Lightning arrow requires a bow, and is therefore affected by that.

If you grab a 'x% cold damage with weapons' then yes, that stat will increase by whatever percent, and will be added to your LA.