Ok, contacted them. Just took off the SD card and saved the playlist directly on the phone. Seems to work without any problems. But I dont have that much space left there and I bought the SD card for the playlists.
OK, it took some time but I think I handled it. I made a clean uninstall, removed the SD card and formatted it with a special SD-Software (free). After that, I put it in again, installed Spotify, downloaed the playlists. After that only one song wasnt playing out of 220. But now even this one works offline. So I guess, with your help, I solved it. Thank you very much!