AC / DC adaptor, unexpected voltmeter reading
That's normal behavior for an unregulated power supply. An unloaded transformer will give a higher voltage than nominal. Putting some load on it will drop the voltage quickly.

For the Arduino I would suggest to use the 9 V setting, so that the Arduino's voltage regulator doesn't get too hot when you would power several things (like a lot of LEDs) from the 5 V supply.