Samsung Galaxy S2 Problems - Possibly Hardware Related?
I am also a user of Samsung Galaxy S2. Mine is rooted but all the indicators of your phone are pointing to a hardware problem. There is most probably a short-circuit like instantaneous electric jumps inside the motherboard. Most of your problems can be answered rationally by uncontrolled electric jumps.

For instance plugging the USB cable (namely a power source) directly starts the phone, indicating that there is a short circuit which transfers the USB power to the power button of the phone whereas they should be insulated.

When you power-off the phone, it restarts. This may be explained with a short-circuit of the phone's battery with the power button.

So on and so forth...