Not sure if my problem, on an iphone 6 with 9.0.2 is similar, but I too started getting the random screen activity. It seems to happen only during phone calls for me. My phone just mutes automatically during the call (the mute button goes on), or sometimes it does the "add call" button just below it during a call, which leads it to open the contact list.
But I figured out that it's not specifically activating those buttons. It's actually randomly doing a pin-point "press" on the screen in a (not perfectly) vertical row area around 1/3 the way in from the left. I have been doing some experiments to prove this. During phone calls, I switch to the notepad and activate the "scribble" function. During the call, you can see a little black dot for each place the phone is self-activating the screen. When it was on the phone call page, it was just muting and adding a call when the dot was in that particular area of the screen. But with notepad you can see all the places it's "touching" during a call, including the "return to call" green bar at the top of the screen which does just that (where it will then likely mute during the call).
I'm now experimenting with the accessibility function where it says "press for x seconds before registering a press." I've made that 0.25 seconds which is not noticeable with human usage, but may be stopping the registration of these sudden, momentary computer-generated touches. Will report back if that's the fix.
The other possiblity for me is I just had Apple fix the bad selfie camera which was moving out from under the screen mask. Maybe they damaged something during the process or gave me a bad touchscreen (did the work mid september around the same time as the 9.0 update. Meanwhile, the new camera is already starting to move too! So maybe I'll have them replace it all again and maybe THAT will be the fix for this. If it is this screen issue, then it's not the iOS9 situation...
