Thank you for pointing me to the diagnostic log. I analysed it myself (it is in German, anyway), found the problem and solved it.
I feel stupid as well as a little bit angry. Fact is, you don’t have to copy avformat-55.dll into the program folder, but ALL THE FILES contained in the ffmpeg zip file (that’s embarassing), but that is mentioned NOWHERE (this makes me a little bit angry).
Anyway, thank you for your fast solution. And to all the folks with a similar problem: download the ffmpeg zip file and extract ALL THE FILES contained within it to the audacity program folder. Then the library needed for aac files will be detected.
In the “full instructions”, it recommends downloading the ffmpeg-win-2.2.2.exe file. When you run that, it automatically installs all of the required files in the right places so that Audacity can find FFmpeg automatically.
At least you are up and running now, and hopefully this thread may help others that have similar problems.