The minimum recommended voltage for the Vin pin is 7V. The Vin pin supplies voltage to the internal 5V regulator and those regulators typically need a couple of extra volts to regulate properly.
That being said: it will probably still work. The Arduino will likely work fine at the reduced voltage.
To turn the motor on and off use an NPN transistor between the - side of the motor and Ground. Connect the + side of the motor directly to the 6V battery. Google for Arduino Transistor to get more information about using a transistor to boost the current and voltage controlled by an output pin.