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.