The maximum voltage at the Vin pin is 12V, so using your battery is fine!
Considering the current consumption:
The arduino itself will usually source less than 200mA, but your whole setup will probably sink more than that.
It doesn't output 0v because it's near impossible. 0.003v is in fact very low. The LOW output, according to the datasheet, can be as high as 0.8v (when powered by 5v) and still be considered valid.
LOW does not mean 0v. It means "below the voltage threshold that is considered a LOW".
No You can easily achieve this by using a NPN Transistor....
simulate this circuit – Schematic created using CircuitLab
You can use BC548 NPN transistor for this is..
But make sure that ground of both emitter and arduino should be same....
Use a power MOSFET transistor. This one from Sparkfun is an example: https://www.sparkfun.com/products/10213
If you use a relay you will probably need transistor and a separate power supply to drive the relay coils anyway, and will also need a flyback diode to protect the control circuit of the relay from the "back EMF" of the relay coils.
DO NOT try to ...