The warning buzzer is supposed to sound when the the key is still in the barrel and you open the drivers door. A warning to remember to remove the key. Yours was sounding with the key removed and the door open. So the door open detection is working but the key in ignition detection has something wrong with it, however that's done? Is there something stuck or missing on the barrel? Perhaps a bad connection or dirt or maybe a fault in the Febi switch.
On most of the radios if the power has been disconnected then the radio will default back to ignition key control.
So yours switching on automatically would seem to fact the facts. I wonder if the failure to auto off is related to the key detect issue.
To check/change the status.
- Ignition on
Turn off radio
Press and hold preset buttons 1 & 3
Continue to press and switch radio on
Continue to press and 'IGN ON' or similar should appear in the display
Continue to hold and release the buttons when you hear a beep
The display should change to 'IGN OFF' or similar
Repeat the procedure to revert to 'IGN ON'
If the radio has had it's power disconnected but doesn't ask for the code it means that the code protection was disabled.
To switch it on again I think you have to follow the usual steps to enter the code. IE ignition on, radio off, press and hold AS button etc.