Conversion (Call od the Brotherhood) question

Hello. I am making lighting / cold arc build and have two question.
Does buffs to lighting dmg apply to converted to cold dmg?
I mean I have a lot of lighting dmg buffs and i am not sure does it buff all of my dmg or just 50% of it.
And second question if lighting buff work for all dmg so does cold buffs apply to converted dmg?
Does it possible to make double buff for this dmg or it's not that funny I thinked is it?

And one question about build. Which unique staff would be the best option? I was thinking about:
Pledge of Hands, Agnerod West, Taryn's Shiver. Maybe there is better one or i should just try to get a good rare staff, but this would be really expensive :/

lightning applies to all the damage yes, including the stuff that was converted to cold. Cold applies to the converted 50% but obviously not the lightning.

I wouldnt use taryns, the other 2 might be good for it. A good rare is always an option, obviously depends on price. When people in my guild have gone lightning with a staff they always seem to be using pledge of hands if thats anything to go by but personally I never really use spell staves so i cant say for sure.

edit: if you really want to make use of cold passives well youd need to use 2 of the rings for total conversion to cold, you can get cold passives, but being only 50% effective it makes it somewhat questionable if its rly worth it when theres lightning, spell, elemental, cast speed and crit passives available that scale all your damage.

1. Yes, modifiers to lightning apply to both kinds of damage, think of it this way: 100 lightning damage, 20% increase -> 120 lightning damage. 50% converted to cold -> 60 lightning, 60 cold. 20% increased cold damage -> 60 lightning, 72 cold.
2. Pledge of hands or duskdawn I'd say, depending on whether you wanna go crit or not.