I have the same multieffects pedal. As you say, it doesn't have an on switch, it turns on when you plug in your guitar in it (or any other cable in the input jack), and will stay on unless you either unplug your guitar, or the power cable.
If you say it erratically turns on and off, I think that means something is not securely connnected with your guitar cable and makes the unit think the cable is being unplugged when it hasn't been, so check it. It can also be a power cable issue as others have said.
As for the fuzzy noise, check first that you are not running any setting on the multieffects pedal that would cause it, like very high gain or running the compressor with high levels of volume. So I'd recommend unselect any effect, set the amp emulation to either clean or off, and see if there is any fuzz. Also on your physical amp, run through a clean channel first.