I've got a Nexus 7 too. Mine just updated itself to Android 4.2.1.
Turning off the toolbar in Coolreader is in the options. On the tab immediately to the right of the one that says "css" there's an option to set where the toolbar should show. I set it to none.
There's also an option to "turn off toolbar on fullscreen", which I"m not sure what it does 'cause so far I don't THINK I can get full screen to work... At least not in what "I" think of as "full screen". No matter what I've done so far I still have a section of screen at the bottom with three dim buttons...
But, given how well the reader works, that's a trivial issue.