Do the mortar damage or auto-mortar advancements affect WZIB ?

There aren't any mortars in WZIB, so no

What are the attacks called that you make against other players?

As far as I know, they are just called attacks. The clan wars reward for taking 25 territories is something like "attacks against you in idle battles reduced by 20% for two weeks."

You know the icon I’m the splash screen for idle next to attacks looks suspiciously like the mortar icon

That's because it technically is the exact same icon. There aren't a lot of icons, and those that exist are reused from time to time. So, calling them mortars isn't that far off. But the attacks in battles are buffed by other advancements/CW rewards/map rewards.

What advancements buff idle battle attacks ?

Reward for finishing Hardened Trikleston level: Protection: decrease attack received on battle by 10%

idle, more like, i want to rub my balls in your face