Good Question, I think it may be possible. On my video iPod, it is possible to listen to jsut the audio of the videos so I don't know what would happen on a Mini
No, I don't believe you would be able to do this. What you could probably do though is extract the audio in iMovie or Quicktime Pro and save it as an MP3 (or AAC or whatever).
I don't know about using a video that you purchased through iTunes, but I have listened to the just the audio from DVD's I've ripped.
If you can't do it this way, just use the method mentioned in a previous thread: https://forums.macrumors.com/threads/189342/