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If you have 2 power supplies, yes... you connect the ground potential of each together to create common ground. It's required to have your circuit "behave" correctly.
Also, couldn't a transistor eliminate the need for diodes?
I'm afraid you lost me here. If you will be driving a 12V MOTOR from the Arduino a diode is of little use. You will NEED a transistor. And you will need the transistor to behave, so driving the base with 5V and no current limiting resistor will "stress" things a bit. Put at least a 1K OHM resistor between the Arduino Pin and the NPN base.
- Rectifiers are for controlling direction of flow. Zeners are for voltage regulation circuits. Zeners do not make good alternatives to rectifiers since they are designed to break down and allow reverse flow above a set threshold... AKA they will behave to be a bad rectifier... That "threshold" is their "regulation voltage" that you see on the package at radio shack. Check out a Zener circuit... you will see them in "backwards" for a reason.