While the other casters pretty much directly harness the power of aether, Arcanist funnel through mathematical formulas to get 'higher' functioning spells. While Black Mages use aether to bring forth fire (Which aether can actually align to naturally, as per Ifrit), the Arcanist renders the stuff in raw power, and then makes constructs with them.,

Bio, miasma, and virus are just physical constructs, like the summons. Think less 'little bacteria running about' and more 'manufacturing nanobots and throwing them in my enemies face.' As such, they can also affect non-biological enemies.

As for the grimoires, they hold all our math in them, presumably so you don't need to remember everything. They don't need to re-write every spell when its cast, but some of the more complicated ones do, not because the act of writing them is important, but because it helps the Arcanist focus.

The books don't fill up because of storyline/game play segregation.

I think that covers everything.