If you are running iOS 5, a new feature actually allows you to do the Wifi sync between multiple computers, just with different types of content.
Your process may be slightly more difficult because you want to have mostly everything sync from a new computer. This will be a lot easier to explain/walk through if you were almost starting from scratch on the iPhone, or just adding a new syncing device. So for some preliminary work, you will at least need to:
- Make sure you are using iOS 5
- Have iTunes 10.5 at least installed
- Make sure you are ok possibly loosing backups. Since this is complicated now (iCloud vs iTunes backup, etc), its really outside the scope.
- Are ok using Wifi Sync
- Make sure that your MacBook Air has the content you want to sync to the device on it.
At this point, im assuming the above has been done.
- On the Mini, plug in the iPhone and make sure Wifi Sync is enabled. Also disable all other content sync except for music.
- Unplug from Mini, and plug into a charger (just to be safe) and test wifi syncing.
- Make sure the sync is fully complete on your Mini.
- On your Air, the iPhone should now show up, as long as they are on the same network.
- Setup sync on the MacBook Air, and select the content that you wish to sync (as in, everything except music).
- Click Apply, so the sync starts.
At that point, you should be done. On the iPhone, if you go to Settings -> General -> iTunes Wifi Sync, it should now show the Mac Mini syncing music, and the MacBook Air syncing apps/photos/etc. I also had to restart my phone for this to show up correctly.
There is a little more information available at http://macmost.com/forum-how-do-i-sync-my-ios-5-device-with-multiple-libraries.html
As for your final question, yes, you could mount your iTunes Music from your Mini on your laptop, but iTunes on the laptop would need to have its own library, and using multiple libraries like this is rather complicated and dangerous. It can easily lead to confusion if you delete something from one library, because it won't be removed from the other.
To share the folder, go to System Preferences, Sharing, and select File Sharing. Then you can either drag the music folder to the first list, or click the '+' to add the folder.