question about skill damage conversion

On the nightblade skill tree Ring of Steel has a skill modifier called Ring of Frost which makes all ring of steel dmg from pierce to cold.There is another modifier called circle of slaughter and it gives pierce % dmg to ring of steel. My question is: does ring of slaughter apply before ring of frost conversion or not? Asking this for other skills aswell.

Yes, see the combat guide for reference, it answers a lot of questions like this.
In specific:

The order of events for Conversion is as follows: Base Skill > Skill Modifiers > Conversion on the Skill or Transmuter > Conversion on Equipment and Buffs > Equipment, Auras and Passives
If you have a skill, such as a Transmuter, that converts the damage your attack does, then this calculation is performed before any bonuses from other skills, attributes or equipment are considered. Skill modifiers are still applied first. What this means is that if you take a skill that normally deals Aether damage and use Conversion to make it deal Chaos damage instead, then the skill will benefit from % Chaos Damage bonuses instead of % Aether Damage bonuses.

thank you very much.