»Direct« as in »a very thin layer around the GPU's capabilities and workings«, instead of what OpenGL does currently with massive amounts of state that the driver has to handle and a movel that doesn't at all work like the GPU does. This removes a quite hefty translation layer between your software and the GPU, leading to better performance, fewer hacks and workarounds (hopefully), etc.

CUDA gives you pretty much full access, and AFAIK you don't need admin priveleges to run a CUDA program. many of my errors when writing cuda code resulted in my web browser being rendered as a texture.

"Direct access" most likely doesn't mean literal direct access, just less convoluted and in the way than current GL is.

I'm unaware of Mantle and Metal, but DirectX is direct too.

It's not really a massive issue when you're locally installing programs since you've already compromised your system in the first place. But when leveraged over the web with APIs such as WebGL, it does become a security vulnerability.

Yes, GPUs these days have an MMU and the idea of multiple contexts in order to separate processes from one another.