Not sure if this is a bug or not, I don’t play Aug, but currently I’m able to get ~300-350k Breath of Aoens procs by bursting it with my Heroic Kharnalex.
I have a tracker for it and the ticks are absolutely being captured by the DoT effect of the on-use.
If it’s intended, pop off Evokers, you will do giga damage with it. If it’s not, well… Enjoy it while you can 
I don’t see why it wouldn’t work the way you describe since Aeons just snapshots all of the damage in its window afaik.
It actually doesn’t. Using damage on use items during the window is a waste, typically.
I’m unsure if the OP’s interaction would be a bug or not. I almost feel like it’s intended since the evoker staff is damage specific to evokers, whereas other on use stuff hasn’t been.
I’d have to play test or but if that’s how it works and it’s intended that’s solid, though probably not enough to make it stronger than lego for those that have access to both.
https://www.warcraftlogs.com/reports/27XkHwnJQgPjGMZF#fight=36&type=damage-done&source=1
Here’s a normal Rashok where I tried it out.
It’s worth noting that when Kharnalex was “used”, Aug didn’t exist, so unless someone was using Kharnalex when Aug was released, this interaction was likely never tested.
I’d say it’s enough to make the staff worth it over the legendary - which would put it back on “mostly pres only or situational” grounds again.
I’d very much doubt it. Getting more personal damage out of breath doesn’t really come close to the higher ilvl for EM and the solid Lego buff uptime.
There would be times in dungeons with burn/prio targets that it probably would be worth, though.