My N7 also has problems. If touches cross the vertical or horizontal axis, the input goes nuts as you see on the videos. The issue is 100% reproducible but goes away if the tablet is plugged in to the USB charger. I have not noticed much of a difference in touch problems if I hold it in different ways. I have also had problems with swiping text via the Android keyboard, and registering simple touches on keys, but only a couple of times.
I observed this before I loaded it up with 3rd party software. When I took it out of the box I immediately applied the OTA update so I can't compare touch performance on the two build numbers.
My Office Max 15-day window is up on Saturday, so unless Google can rush out an OTA patch by Friday I will have to return it. I'm hopeful that this can be fixed with software, but if it can't I don't want to incur the delay and expense of shipping it out for warranty service. It'll be much easier to just buy a new one when it is safe to do so.