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!

Sort of depends on your tree honestly. I've done 2 different ice strike builds now, and realized quickly that it doesn't matter if you go ele or phys, as long as you focus there. If you spread it out you essentially get DMG from nowhere rather than both (this is a QOL lesson for any build really).

If you want to take advantage of the phys conversion, and ice strike build could possibly be among the most expensive builds lately game due to a top tier phys DMG quarterstaff, with the attk spd and crit/crit multi you will need, is upwards of hundreds of divines in this current economy and even more expensive to craft.

Ele DMG is definitely more accessible because of the passive tree, and if you can build a decent amount of added AOE from jewels, anoints (merc has AOE bonus notables), etc. it can be just as strong as using the phys conversion.

For late game mapping on an ele DMG build try running overwhelming presence (I think that's the name) - I love it on my farming build, which is a quarterstaff build. If reduces monster ele resist by a LOT when it's maxed.

Let me know if you have any other questions or I didn't answer something, happy to help where I can.

Just for some cost perspective, I built a HOWA/Regalia/Ingenuity/Pillar build for chepper than just a 700+ staff with 2 attk/sec and decent crit and crit multi would cost, both on the market and to craft.

For accessibility, go ele DMG and just get a decent mid phys staff with a bunch of crit and spd, ele DMG and build flat cold and lightning on your gloves and rings.