
There have honestly been a lot of arcanima threads lately and I'm sure someone will link them/quote from them soon but right now I'm just gonna focus on these specific questions. Some of this is confirmed ingame but we don't know many specifics of arcanima at all and a lot of it is theoretical, headcanons, educated guesses, etc.
1) How does Arcanist magic work? I know that they use mathematical equations related to the arrangement of different symbols and whatnot to cast spells (like badass geometry?
), but do they have to carve the symbols onto something? Do they have to touch a diagram with their palm? Do they have to say something related to the diagram? Or do they just have to… I dunno, look at the diagram?
Arcanima is shaping aether through geometry into spells. They cast them using their grimoires as a focus, and the grimoires contain geometries and glyphs written with special aether-conducting ink.
2) What purpose does the grimoire actually serve? Is it just a handicap, so they can have a visual of the symbols to help them? I.e., can they cast without the grimoire, and it’s just more difficult?
See above. It's a focus. I think it could be possible for an arcanist to shape magic without one if they knew the diagrams very well and could manipulate aether precisely with just the hands, but a lot of people think of the glyphs in the grimoires as circuits for the aether to travel through if that helps you imagine it.
3) Speaking of carving symbols into things, I read to summon carbuncle they have to carve the summoning symbols into a crystal. Is this necessary? Are the crystals one-time use? Why is this the process? I.e., is the necessary aether stored in the crystals and used up once cast?
The game never goes into this afaik. My headcanon is that yes, you need one, but you don't have to carve stuff INTO the crystal (the "carbuncle's ruby"), but it's the cut and size of the gem itself that shapes the carbuncle you summon with it. They don't get consumed. Channeling aether through the crystal is like filtering light through a prism or something. Aether goes in, carbuncle comes out. It is the process because it's Final Fantasy.
4) Do arcanists use solely their own aether, or do they also draw from other things? If other things, what other things? Do they have to be living? Do they have to be crystals?
They use their own aether and they channel it through the diagrams. I think they could use crystals if they wanted to, though. You can be creative with this.
5) What purpose do carbuncle serve? Are they mostly for extra firepower? Does it damage the arcanist when their carbuncle is hurt or destroyed? Is it the same carbuncle each summon?
Mealvaan's Gate, the Arcanist's Guild, is the also the customs office of Limsa Lominsa. Carbuncles, I imagine, are used like drug dogs and the like. You could get creative with this since they seem to have some sentience to them, like a dog or something. The carbuncle being hurt doesn't hurt the user, it's a different entity. I like to think each crystal's carbuncle is unique and could be the same one with each summoning, especially since downstairs int he Arcanist's Guild they appear to be training carbuncles. if your carbuncle couldn't retain memories this would be useless, unless they're just practicing controlling carbuncles and not actually training the carbuncles.
1) How does Arcanist magic work? I know that they use mathematical equations related to the arrangement of different symbols and whatnot to cast spells (like badass geometry?

Arcanima is shaping aether through geometry into spells. They cast them using their grimoires as a focus, and the grimoires contain geometries and glyphs written with special aether-conducting ink.
2) What purpose does the grimoire actually serve? Is it just a handicap, so they can have a visual of the symbols to help them? I.e., can they cast without the grimoire, and it’s just more difficult?
See above. It's a focus. I think it could be possible for an arcanist to shape magic without one if they knew the diagrams very well and could manipulate aether precisely with just the hands, but a lot of people think of the glyphs in the grimoires as circuits for the aether to travel through if that helps you imagine it.
3) Speaking of carving symbols into things, I read to summon carbuncle they have to carve the summoning symbols into a crystal. Is this necessary? Are the crystals one-time use? Why is this the process? I.e., is the necessary aether stored in the crystals and used up once cast?
The game never goes into this afaik. My headcanon is that yes, you need one, but you don't have to carve stuff INTO the crystal (the "carbuncle's ruby"), but it's the cut and size of the gem itself that shapes the carbuncle you summon with it. They don't get consumed. Channeling aether through the crystal is like filtering light through a prism or something. Aether goes in, carbuncle comes out. It is the process because it's Final Fantasy.
4) Do arcanists use solely their own aether, or do they also draw from other things? If other things, what other things? Do they have to be living? Do they have to be crystals?
They use their own aether and they channel it through the diagrams. I think they could use crystals if they wanted to, though. You can be creative with this.
5) What purpose do carbuncle serve? Are they mostly for extra firepower? Does it damage the arcanist when their carbuncle is hurt or destroyed? Is it the same carbuncle each summon?
Mealvaan's Gate, the Arcanist's Guild, is the also the customs office of Limsa Lominsa. Carbuncles, I imagine, are used like drug dogs and the like. You could get creative with this since they seem to have some sentience to them, like a dog or something. The carbuncle being hurt doesn't hurt the user, it's a different entity. I like to think each crystal's carbuncle is unique and could be the same one with each summoning, especially since downstairs int he Arcanist's Guild they appear to be training carbuncles. if your carbuncle couldn't retain memories this would be useless, unless they're just practicing controlling carbuncles and not actually training the carbuncles.