Not sure if this will help much, but even when plugged in via USB my computer will not recognize the device. The S3 states that it is connected and it does show up in Device Manager, it just won't show up as any kind of readable memory.
Trying to ADB sideload a rooted stock 4.3 ROM now. Any advice at all would be a help.
EDIT: I took a video (https://www.youtube.com/watch?v=tJ-eaQ_9mVI
) of the problem so everyone can see what I'm talking about and I also found a fix.
I'm not sure what was causing it, but after a solid day of researching it online the closest thing I got was resetting app preferences and also trying to clear any data of Google apps. This seemed to sometimes slow the popups down for me or create a longer gap before they just came right back in full force. So, going off that small success I had managed to navigate my way over to Settings -> Accounts -> Backup and reset -> Factory data reset and ran that. The first boot after that took a very long time, I'd say a solid twenty minutes but it did work. No more error and everything has been going smooth for me since. A restore of apps with Titanium Backup also did not bring the problem back up again, so the device seems to be good as new again.
I apologize if any of these is out of line, I just thought I would follow up with a solution in case anyone else has this problem and can't seem to fix it. Maybe this will help them.