On my board all of the ground buses are connected and I use separate buses for different voltages, so I'm (generally) automatically good on the grounds being connected. (I am powering other things directly from the Arduino so the ground is already present on the breadboard.)
Would a TIP120 work with the 5V from the Arduino?
To be honest, I don't have the electromagnet in my hands because I have yet to make it. I am going to try and wind magnet wire around the core of an AC induction motor out of a microwave, which I should get tomorrow. I was playing with a "test" electromagnet today though, and off of a 9 volt it drew 1.2ish Amps. So...I'm hoping I can play with these numbers a little and say that 9V/1.2A = 7.5? in the system. Provided that is correct, 7.2V/7.5? = 0.96A. I also would like to believe that making the actual electromagnet will require more wire than the beefy nail I made today, so the resistance will be higher, and thus the current lower.
I realize that these numbers are pretty low... If you don't mind me bouncing another idea off of you: What if I wired 2-3 9 volts in parallel to give this a little extra current capacity, put some resistors in parallel with the electromagnet to reduce resistance, thus causing more of a draw on current? Do you think this would be viable for a short period of time? I really only need this to work for about 15 minutes tops.
Also, this just came to mind: Because an electromagnet builds up an inductive field, should I be putting a diode backwards across the inputs of the electromagnet to protect the transistor from the potential voltage spike when it is turned off?