Something is definitely wrong with the iPad (but, you knew that). I would suggest to try a reset of the iPad to start. To do (apologies if you know how ... or already have):
Reset: Hold both the power and home buttons until the screen goes dark and the Apple logo then appears (the slide to unlock screen may or may not appear). Once the logo appears, release the buttons. After a time, your homescreen will appear. Please note that it may take some time for the Apple logo to appear or for your homescreen to appear (or both). This is normal. Eventually, the homescreen will appear.
If that doesn't clear it up, I'm afraid you've got more wrong with the iPad than we (or software) can fix. Then, it's time for a trip to Apple and see if the Geniuses there can help (or replace it).
Good luck and sorry for your troubles.