Update on my saga:
The apple store wiped my Mac Pro and handed it back with a freshly installed OS. I loaded up premjere CC 2014 and edited on it for a few days, then finally had to render something yesterday. The random green lines appeared almost immediately. This eliminates software from my possible issues.
My apple store is now going to replace my graphics cards. If this fixes the issue, it's been a matter of me having bad cards all along. If not, they will request a new computer for me.
Until next time...