Analog input draws current when Arduino is turned off
Yes, it is normal behaviour, and it's called back-powering.

The current is flowing into the input pin and then flowing up through the ESD protection diode inside the chip to VCC.  It's bad, because the ESD protection diode can't take much current and is easy to damage.







simulate this circuit &ndash; Schematic created using CircuitLab

You can prevent it from happening by adding some isolation into the input:







simulate this circuit

MOSFET M1 will be ON when VCC has power, but off when it doesn't. The 100kΩ resistor pulls the gate to ground when VCC is off so it doesn't float. The diode is the body diode of the MOSFET added for clarity. You don't need to add a diode there.