Encountered similar issues with my newly arrived Fairphone: touchscreen going wild, registering ‘ghost’ touches, calibration being way off. Then, disconnecting the phone from the USB charging cable made it all disappear. Thanks to Malte for the suggestion! Got a smoothly working phone now 
Onno
I have created your post as a new topic as this is a different problem. Some chargers have been known to cause this issue on the FP (and some other models of phones from Google searches). It is usually caused by unshielded chargers causing radio interference or from placing your phone on a surface that may make the issue worse. One guy reported on XDA that placing his phone on his bed caused this issue which was probably due to RF interference amplifies by the metal springs of the bed
Usually trying a different charger or cable will resolve the issue. Also charging from a computer usually doesn’t cause this issue 
Ah, OK, thanks for cleaning up my post, and for the additional info I find it not to be a big deal, once you know what causes it. And indeed, charging from a computer or laptop does not seem to cause any issues.
Cheers!
This happens to me too. I found out when you consciously touch the metal back of your FP, the touchscreen works fine.
Another discovery: When my no-name-charger is plugged in, my radio set begins to hiss and it sounds like a bad radio signal.
I moved a post to an existing topic: Buttons pressed randomly