Samsung Galaxy S3 keeps restarting, stuck on Samsung logo
It does look like a stuck power button. (as ridiculous as it may sound)

The behaviour in your phone is exactly as it is when I press and hold on my power button for an extended period.

By right, the device should not turn on by itself when the battery is inserted. It should remain turned off - only a hardware switch can turn it on. As your phone turns on by itself, it clearly shows that the power button is being pressed.

Furthermore, the power menu shows when you turn the phone on without touching the phone at all and again showing the power button being pressed.

I suggest you bring your phone down to someone who knows how to open up your phone to see if there has been any damage to the power button.