Catalyst for minion build

I am using the unique belt Darkness Enthroned with two jewels which have life modifier and minions deal additional cold damage modifier.
If I use Turbulent catalyst to improve the quality of this belt, will it increase the additional cold damage?
I don't know much about viewing stats of the minions so not sure about this.
Another question is if using Fertile catalyst will enhance the life modifier as it is on the jewel not on the belt.
Thanks.

No. Catalysts only raise the relevant stats on the item itself. No catalyst will do anything for Darkness Enthroned.