Slitherine

would EAP Immortals be uniform - or would robes vary with regiment or individual?

I've heard arguments for both. You'll have to decide for yourself.
Osprey has an interesting book on Achaemenid Persian armies. Try that source.

I've painted my Later Persian foot with a unit-based tunic and shield, but with their own trousers etc.
My Guard Cavalry and Foot Guards will be uniformed in the royal pattern -- same as Darius.
Prestige units will be uniformed, others not so much.