Using this ring on a Deadeye with lightning build (new char, level 35)

How does it work with lightning passives (shock/chain lightning) and extra lightning damage?
Is the calculation done for lightning damage beforehand, then damage is converted to cold afterwards?
Using this logic, do I gain all the bonuses of lightning damage but it freezes enemies on top of it? Or does converted damage lose extra damage to shocked monsters you would have without the ring?

As I have understood how the ring works is that; All the lightning damage done will be converted to cold the moment it hits the enemies, and not afterwards.

So yes, you do gain all the bonuses of lightning damage converted to cold damage, meaning you will freeze without issues with the ring equipped. Extra damage to shocked monsters will be nullified since you won't deal any lightning damge, as far as I've understood.

Someone else correct me if I am wrong here please, don't mean to misslead you with false information,- but this is how I've understood how the ring works and also after trying it myself on my LA ranger.

Hold alt button and click on converted keyword in ring description
It will explain how conversion works