And Enjolras282 got it right it's the rising sun, most of the reasoning is right too...

Some from the schizofrenic, - Saw blade is needed from Belhelmel.
and some from the weakest, - Screws are needed from a Geezaur.
but that's not all... you also need gil

Something that Fujin could use, - Is a type of Razor Ring.
and yet we see it every day. - We see the Sun rise everyday.

I'm a veteran, rising sun was in ff7 too if i'm not mistaken
going round and round... it spins when flying
flying like an arrow in the sky... it's a projectile weapon