Enable Disk Use in the iPod Preferences. I don't recommend moving apps outside of the Applications folder however feel free to drag other files. Apps aren't generally very large anyway plus the iPod has a slow, portable hard drive that shouldn't be overworked. If you have Garageband, move the loops to the iPod.
I've never moved my loops but if Garageband doesn't let you manually find them (which it should) then you can always keep them on the iPod and temporarily put them back on the mini when needed.
Try Monolingual to get rid of many unwanted language files too.
As far as I know, no. When you plug your iPod in (when it has files on it), it behaves the same as a flash drive/memory stick etc. You have to go into the disk to active/use any of the files/apps on it. If space is becoming a big issue, perhaps you should look into getting an external hard drive.