So I just got my Nexus (LTE) and having the same problem (connected to car's stereo, and get this static noise WHEN no audio is going out, and even worse when the screen is OFF)
It's fine when music is playing, but really annoying if I have to pause it.

OP, Have you found a fix yet? I was also wondering if every Nexus 7 is the same way or there are some defective models.