In which case how many volts would that take.
You seem to be obsessed with volts where as it is current you need to worry about.
Can you post a link to this motor driver you have?
Also would it be wise to use the same battery to run the Arduino.
Yes.
But do you have any decoupling in the wiring?
See http://www.thebox.myzen.co.uk/Tutorial/De-coupling.html
Electromagnets do not generate nearly as much interference as motors do, but you get some spikes on switch on and off that might cause you trouble.
Exactly how much magnetic field do you need from your electro magnets?