So first of all.
“Danke . Das Video war super und du hast mir sehr geholfen. :)”
Which means something like: “Thanks for the great video. You helped me a lot!” 
I could directly find out what my problem is. It was some setting in Blender itself.
In the screenshot you can see the standard configuration of Blender for Units.
In standard “NONE” is selected and this seems also to be the correct setting.
In several “perhaps older” videos on YouTube they recommended the metric system with centimeter selected and I had this setup.
I already thought about this setting some days ago but forgot it and so I didn’t try it out.
Your video showed me the scaling setting (scale 100) which is some really good idea because it seems to work better than switching to centimeters.
It looks like that something (perhaps the skeleton or some other parameter) doesn’t scale with the UNITS settings of Blender when exporting.
I don’t know if this is a bug or if it is intended like this because everything else scales nicely with the centimeter unit setting.
Thanks again for your great help. I never thought that someone is so friendly and answers with a half an hour video. That’s awesome. 
P.S.: I could also get rid of the smoothing groups error by exporting it with Smoothing “face” in Blender.