Since you did a cold start (shutting down then back up) have you tried the Apple recovery? At start up hold the Command-R at bootup. Then let it go through the stages of that. Once it shuts down and reopens go to your app store and download a fresh copy of your Mountain Lion software has installed it should stop the freezes.
Keep in mind that another reason it could freeze is that you have too many apps running at the same time. You could do a factory restart (once you update your icloud with everything that is on your mini ipad.