What if you press the power button and hold it for 10 seconds?
If it turns off, the power switch may be configured wrong.. there's an option in the Control Panel / Power Options to configure what the power button does when you press it.. it can either go to sleep, go to standby, shut down or do nothing. This should not affect the computer when Windows isn't even loaded, but it just may affect it if you use sleep mode instead of shutting the computer down.
What you describe (powering up as you connect AC) can also happen if the PSU is damaged.