Oh yeah, I also had the issue where it wouldn't stay turned off. This is because, as has been said about a million times now in this thread, it thinks its in a dock. This is the same reason the volume slider disapears.
When this problem first surfaced on my phone it was nowhere near as widely known as it is now. Being a somewhat savy user I tried EVERYTHING these people are suggesting the very morning it happened to me as did a friend of mine with an iPad 2. Cleaning the connector, resetting, restoring, its highly unlikely any of this would work because its a software issue. You can go get a new phone, and yeah, that will fix it, but only because the bug didn't manifest itself for whatever reason on the new phone. The problem is the software. THE PROBLEM IS THE SOFTWARE. If you pay $200+ for a new phone now its kind of a waste of money since it will be fixed as soon as iOS gets updated.
BTW, as an update to my situation, my phones has been working perfectly for the past few days...no idea what happned since Apple, amazingly, still hasn't shipped 6.01+ yet. The only things that has changed is 3rd part apps being updated so its possible that the problem does actually result from a combination of background apps and the buggy iOS6 itself. I'm not sure how that could be the case though since I force killed every app, rebooted, and restored the phone day one and had the issue still.