Got a micro-USB-cable, updated the firmware, see the video, and tried an other remote again. In Test-mode, while teaching the new remote, all three buttons always(!) work, as soon I leave to normal mode. The new remote the same as the original remote.
The problem still exists. Turning the game frame on, out of “stand by” always works(!), but it worked befor updating the firmware too. I still have to press the buttons 20-40 times till game frame reacts, no matter which button.