Thank you all for the helpful tips. @ThePara, it turns out in this case the GPS issue was the telemetry radio strongly interfering with our GPS. We discovered this by chance, where we found that the GPS would work correctly, reliably, and with very high precision when connecting the ground station computer to the pixhawk over USB (thereby reducing the data being sent wirelessly). Repositioning the telemetry transmitter completely resolved the issue. As a backup and for improved accuracy we have also added a second GPS module and configured GPS blending, the drone now carries a Here+ (primary), and a Here2. We have managed to do some flights (see Tuning 1.4m X8, oversensitive including movies). The compasses are still quite unhappy, so we are going to place the Here modules further away from any large current wiring, but the drone is flyable even though it will require some tuning.
@Nolan_Reep, @cyberpilot360: thanks for the tips, we checked it out: It seems that the ESC calibrations were very good.We found that for all motors that the throttle percentage to make them turn where all between 5 and 6%.
@Eosbandi: In the end we did not have the nerve to carry the drone above our heads. We checked and checked again the motor responses to RC inputs and tilting the drone without props. In the end it turned out to be correctly configured, but the stock PID values did not control the drone very nicely (see also our other post linked above) which may have given the unstable appearance during takeoff. We figure it is best to take off confidently, and not linger too long with legs barely touching the ground but it takes some courage to do so.
Once again thanks for all the help on getting this thing in the air. We still have some tuning questions in the other post, which we hope to resolve.