Unbelievable, but my problem is gone with a simple fix! I was suspecting it was something mechanical - the screen was acting erratic, as long as the phone was lying flat. As soon as I touch it with the second hand, or take it up into my hand, the strange behavior was gone (most of the times). But now I have bought a simple plastic protection case, and all the weird behavior is gone completely! I have no idea what happened - maybe it has something to do with grounding, but I finally can use my iPhone again.
Old thread, but I found this because I have exactly the same problem. Apparently the display port is soldered to the mother board, so you need strong technical skills to do it yourself. My guess that most technicians would advise just changing the logic board :/