Thanks again for your answers.
So what I've been trying now was to borrow a Raspberry Pi from a friend since I don't know anyone with an Arduino. I connected the gyro to the Raspberry and it showed life with a shining LED just like when I connect it to my Arduino. But just as with my Arduino it's still not working. I run a I²C scanner on the Raspberry but it couldn't found it, too.
Assuming that I did everything correct it seems like that it is at least not a problem of my Arduino. So to keep the search going I decided to try the cheap version one more time and I ordered another module in the internet from a different shop this time. Additionally I also orderer a simple I²C display so that I can try another I²C module in case it should still not be working. Maybe it's still just me and I am too stupid to get it working or something...
I will see and report again. The pieces should arrive during next week.