Fascinating!
It's not the third-party keyboards. I uninstalled them, and it's still Ghost tapping AND typing.
But! I THINK it's the charger. Or charging. I noticed it only seems to happen while charging. If I unplug it, it'll work fine again. And seems to keep working. I plug it back in, and within a few minutes it acts up again. Repeat.
It's bizarre. Without touching a thing, I watched it type something like "skskskssssssssksksksksksks." Then it opened Control Center. Than the Notifications drop-down. It's favorite thing is if I tap a Message text, it opens Find Phone. It selects Mails. I also sometimes have to swipe 5-10 times to get past the Lock Screen.
Maybe it's a preview of iOS9's new Proactive feature?
I joke.
Seriously - so how the heck could a faulty(?) charger cause an iPhone to tap and type random things?
Assuming that's what it is. I'm not 100% confident, but so far it's been consistent: charger on, it eventually Ghost taps. Charger off, it works again. I experiement with phones quite a bit, but I've never seen anything like it.
EDIT: Ha! It's not Just Me
https://discussions.apple.com/thread/5956716
Beware of non-OEM chargers!