Use a single power suppy (GND and +12V) and tie the Arduino’s GND with that GND.
Your circuit fails because the Arduino’s GND is tied to the -12V from the dual power supply, so the +5V that the Arduino outputs equal to -7V on the non-inverting input. Since this is a non-invering schematic, the opamp doesn’t invert the signal. Instead, it tries to double the -7V to get -14 on the Out, but since you’re powering the opamp with -12V, it can’t achieve a voltage that low, so it outputs the maximum it can give: -12V.
The LED turning on even when there’s no signal on the non-inverting input is probably a floating input problem. It picks up EMI so it just amplifies that. Try connecting the non-inverting input to GND, the LED should turn off… that or you burnt one of the opamps, lol, try the other one in the package.