If this has already been confirmed either way, then I must have missed it and I'd love to know more about it. So, the central question here:

Are typical Soul Gems made of Aetherium? Are they made of the same material as Welkynd and Varla Stones?

As I've seen, soul gems come in three basic flavors. Classical (five levels), black, and Azura's Star. A large capacity classical gem can be turned into a black soul gem with the proper rituals. Azura's Stars are manufactured from Azura's own blood (as suggested by Blood of Daedra in Oblivion). In Skyrim, you can then tamper with a star of Azura and make it black. The classical gems can be found in Skyrim when mining geodes in Blackreach and when cleansing the Mana Wells for the Drevis in Winterhold (with those mysterious tuning gloves).

Aetherium Wars tells us that Blackreach was essentially a massive Atherium mining scheme, and I assume that Aetherium is a primary component in those semi-translucent blue rucks jutting from the ceiling.

I could ask a bunch of followup questions (what is Aetherium, is the Eye of Magnus a metallic Aetherium shell with a magical core, is a black soul gem some mix of Aetherium and Ebony????), but I do have one question that I wanted to ask while I was here and it's related to all of this: When the builders of Nirn, with Magnus as their architect, fled into Aetherius and tore holes in Nirn (forming the sun and the stars), did they all exit through the sky? The sky is the edge of Nirn, I assume, but can't you mostly just poke through reality at any point and escape from it? The portals out of Nirn (Sovngarde, Nocturnal's realm, Soul Cairne) are all placed wherever they feel like. Could the Eye of Magnus be a tear into Aetherius that has been sealed? Could there be others like it hidden deep within Tamriel that have been sealed? That's a tangential question, and more than a few steps away from what I'm asking in the first place, but nonetheless it is something I am curious about.

I can tell you right off the bat that Soul Gems != Aetherium. While Soul Gems can be mined in Blackreach from the more ore-looking spots, Aetherium is the huge glowing stuff that you can't mine. "The Aetherium Wars" states that Aetherium could not be mined normally, and required very precise tools to do so. Tonal tools IIRC. And on top of that, the only forge that could work Aetherium was more like a volcano than a forge. As for your other questions, I don't know.

I have a theory that the Aetherium is what twisted the Falmer.

I think KINMUNE is/was/will-be a mining tool as well.

Does it have anything to do with Tonal Torture? Maybe the Tones resonating from the Aetherium drove them mad? Or something? Apparently even small shards of the stuff had to be kept out of the cities because they were "harmonically volatile."

As much as I dislike talking about KINMUNE (for personal reasons) that actually makes a good amount of sense at first thought.