Well done. You got there in the end. The lambda sensor needs replacing. Plenty in the breakers!
The solenoid is controlled by the ECU and is dependent on the throttle position. It should only be open when the throttle is pressed not when idling. Have you got a meter? What is the voltage doing in the plug to the valve? Disconnect it and push both probes in and see what the readings are. If they are static at either 12 volts (or possibly 5 volts) then the valve is faulty. If the voltage is fluctuating rapidly then try and do an adaption on the throttle with the diagnostic tool. Otherwise maybe best leave it unplugged.