Yo people, I got a similar problem, with a few charming and bizarre variations.
So the issue is like you all describe. It's erratic, types random $@$*, and a lot of the time doesn't allow me to interact with the phone properly. But it comes and goes - it's not as bad for me as for others in this thread.
I thought it started cause I charged my phone in the USA (I'm from Australia) because it begun the first morning I left my phone plugged in. But that can't be it - a mate has charged his phone from the same charger/adaptor and his is fine. Also, having read up on the iphone power adaptor, it can't be that I've &&^&@@ up the battery with too much voltage.
That said, the phone was pretty hot... that was weird.
Anyway, the first day this happened I took the phone to a phone doctor. Annoyingly the phone was acting almost normally by the time I got there, except that the bottom centimetre or so of the screen didn't work. So the dude replaced the screen. Buh-bye $100.
There was a definite improvement with that, and the bottom of the screen has "worked" since then. But the phone still goes crazy. And the weird thing is, it does it much more in the morning! Like, when I wake up its barely useable. Gets better but still frustrating over breakfast, and by lunchtime it's more or less normal.
Its bloody weird. The other thing I've noticed is a very fain pattern of long grey rectangles on the screen. Like, everything is visible just fine, but there are these faint closely spaced rectangles/lines in the background.
I've done a soft reset and a hard reset, neither did $@$* for me. I can't delete any exta apps cause I don't have many installed, and I use everything I've got. I mean, the phone is *mostly* functional, so I dunno. But it's &&^&^$^ annoying.