Thank you everyone for your help.
I have tried some of the things that were new and will get back to some of them but first if all, I decided to install Piplay on my laptop as well to see if it made any difference. It's a Lenovo Yoga 520 with a Geforce 940MX, so it is below specs and it is NOT possible to deactivate the onboard GPU since it is a necessary feature for converting the laptop to a tablet. (You can deactivate the Geforce though, but that doesn't really help.) I decided to try anyway.
Basically the result was the same on the 360 video. Eyes are looking in different directions and it's a general mess. The game however was much better. I was able to start it (unless I missed something, it's actually just a game gfx demo as you can't actually do anything apart from looking around you) and though it is obviously on a too low framerate (it also seems to play in slow motion, but I think that is the way it is supposed to be) at least it worked.
This makes me think/hope that the 360 movie is messed up but the HMD works, so let's get back to my gaming computer.
I have reinstalled Pimax several times by now, always as admin with the AV off and after deinstalling Piplay, Dr.Pi and the Klite Codec Pack. Last time I even deleted the Pimax folder and deleted every single registry entry containing Pimax or Piplay. None of this has helped, so I don't think this is the problem.
I have deactivated the onboard graphics to the best of my knowledge and it is not visible in the Device Manager. Last time I even activated it in BIOS, then deactivated it in the Device Manager before deactivating it in BIOS again. Also, Dr.Pi detects the Geforce card as GPU and nothing else and checks the specs as OK for 4K. If this is still the problem, I don't know what to do. The only thing I can think of is to update the BIOS which I may try later on.
@Enopho : "If you want to see what it's like, download Unity Chan for a pretty good demo"
- Where? The only things available on Piplay are the three items I have already tried and on Steam I can only find "Attack of the Giant Zombie vs. Unity Chan", but not for free.
@Heliosurge, thanks. I'll try the 2.0.7 later. (downloading but have to go to work)
@PM_Sean, thanks. I managed to update the firmware with your help, but unfortunately with no change.
What I am still thinking about are the drivers. Not the Firmware or Piplay, but there must be some kind of drivers installed for the USB connection and possibly HDMI. Any clues? I have deinstalled both the Audio and Video component in the Device Manager, but when I reconnect the HMD, it's back up again. Obviously deinstalling the devices doesn't delete the drivers.