I've been experiencing the whole spectrum of issues reported here with the 2013 Mac Pro and Premiere. I do think the freezing/lockups are related to the colored lines, screen artifacts. Every time I start seeing the screen issues, within a matter of minutes the machine completely locks up, like back in the old days before OS X. Nothing short of a forced shutdown will bring the machine back to working order.
I've also experienced system-wide lockups just by having Premiere running in the background. I left it open last night, machine went to sleep, I came back later to wake it up and found that the mac had frozen up during sleep!
Jason, I don't think the screen issues are related to any typical video compression or interlacing issues. I have experience dealing with those issues and believe this is not related. It does however remind me of when my 2008 Mac Pro graphics card was failing. I would get similar screen weirdness splattered across the entire screen. Replacing the graphics card fixed this issue on my 2008 Mac Pro. What's curious about the current problem is that the screen artifacts are usually contained within the preview or program monitors, which suggests to me a software issue. I think we can corroborate this by some reports I've seen of users getting their graphics card replaced by Apple and still having the issues.
Regarding the OpenCL discussion: Users of the beta of OS 10.9.4 are reporting no remedy for these issues. Do we know for certain that the beta of OS 10.9.4 contains the latest version of OpenCL? I think it's supposed to be version 2.0
I'm hoping we hear an official word on this soon, as my pricey new machine isn't good for 90% of what I need it for. Using my 2008 Mac Pro to edit video for now, feeling like I'm back on my Commodore VIC20.