Actually I should amend my snarky post.
Here were my symptoms (were, as in I think I just fixed it). I have an iPad3
Could not power down... It would just power back on all by itself.
Could not watch videos in any player - they would pause and/or the audio would stop.
Key clicks were intermittent... They would come and go as I typed.
So I did what several people suggested and I got a small brush like the kind you get with a coffee grinder or an electric razor and I cleaned the docking connector with rubbing alcohol. Really clean... Obsessively clean. That was ten minutes ago and it has been perfect ever since. I have not dropped so much as a single key click in this entire message. You tubes and videos are playing completely normally. I powered it off and it stayed off.
Trust me... Just clean the ****** connector. It only took me a few months to give in. Duh!