Priority on quarterstaff. Ice strike invoker monk.

Invoker monk building into cold damage. Looking to upgrade my quarterstaff and wondering which stats to focus on. I have rings with about 40% increased cold damage and some jewels with elemental damage. My questions:
1. Does it matters if the staff has more flat physical damage and low % increase versus the opposite being low flat physical and high % increase? If the calculated damage (shown on staff) is the same, when does it matter?
2. As ice strike converts physical damage into cold, would it not be of utmost importance to have high % increased cold damage on staff? Or is the % increased cold damage not applied to the converted physical damage (ie. just flat cold on staff is increased?).

So my thinking is:
Phys damage -> cold damage -> % cold damage increase from staff -> % cold damage from rings/passive tree.

Could someone confirm or correct me, maybe with an example if i got it totally wrong. Which stats in what order and amplitude should be prioritized?

Many thanks!