I just got my phone as well probably the same batch on the 5th of august. I have the exact same issue with the display. Random touches everywhere. It disappears after you lock and unlock the phone a couple of times. But it gets irritating after a while. Please update this thread as soon as you get a reply.
I have quite similar situation ONLY when I'm charging the mi3: the touch screen appears completely crazy and touching the screen to unlock the phone or to click upon an app become really difficult. It's like if a "ghost touch" would be upon the screen. If the phone is not charging on the wall, no problem: never had a situation like in the video upon. I don't think it's software problem, because I've installed all the possible Roms and the situation never changed. But you can try to install another ROM.