Hatred/Herald of Ash Elemental Interaction

I want to build a physical bow build with Raider and run Hatred and Ash to scale off the physical damage. If I take Avatar of the veil, does the 40% increased elemental damage increased the fire and cold damage provided by hatred and ash?

Thanks!

Oh! and added fire! any other good things to scale this type of build with?

Yes. Both "Extra" and "Added" elemental damage will be boosted by Avatar of the Veil (As well as any other Elemental damage modifiers)

As far as scaling a build like that...

Doomfletch's Prism springs to mind, alongside a bunch of Flat Physical damage (Doomfletch works well off flat physical, Hatred + HoA just want to you to have high physical damage)

The Signal Fire would also grant some Extra Fire damage (In addition to 50% conversion to fire and some Added Fire damage)

Finally, Taste of Hate and Atziri's Promise for even more Extra Damage (Cold and Chaos respectively)