They don't really explain the giant sphere but I believe it is related to each Exo's memories. Randomly, you will get a voiceline from Ghost when you reach that area if you are an Exo, talking about how your memories are probably somewhere in that area. Clovis Bray, or more commonly called Braytech, was a massive corporation during the Golden Age. It had facilities on most planets and was in charge of the Warmind project, which also can be found throughout the Sol System. They even had cities created by them using SIVA technology on Mars and Europa at a minimum. They absolutely had all the money for this and the promise of living past 300 years was certainly appealing, for the survivability of Humanity. However Clovis had his own nefarious plans to be the god of Humanity and the LUCA of his own race of Humanity by way of the Exo project. In the end, he couldn't realize his plans due to both Ana's meddling with Rasputin and the Collapse crashing down before completion of such projects. If you haven't already, have a look through the Perdition Lost Sector. The top area where you fight the boss is windowed to experiment observation rooms, where you can clearly see their trials with Exos and Vex. Radiolaria always won in the end and had to be destroyed before becoming fully consumed, even Asher Mir struggled with what Light he had remaining to stop it from spreading from his Arm, at least if I am remembering that correctly. The Conductor has more control over the Vex and should theoretically be able to prevent a full conversion from happening, just enough to work for her purposes I believe. It is very possible that we could get new enemies from her at some point but who knows? I absolutely appreciate more of the lore surrounding Braytech, but most of that story has played out, especially with the end of Season of the Seraph (which I still feel like they mishandled) so it is unlikely that we will get more but one can always hope. If it was still around, I would have recommended to check out the Braytech facility on Mars which had more information on the Exo and Warmind projects, though Beyond Light did most of the detail on the Exo project itself, explaining how they were created.