This is an old type motor driver, and there's nothing in your description that tells me you need it. A simple MOSFET as in the original schematic will do just fine to switch on/off a motor, or even to control its speed (using PWM).
If you really need more control over your motor such as changing direction as well, there are better motor drivers around. MOSFET based so less voltage/power loss in the driver, and much less heat.
Servos are not normally connected to a motor shield, the control wire normally goes directly to an Arduino pin. Make sure yours actually need 12V. Check the power rating of your conveyor motor as well, make sure whatever motor driver you use can handle the stall current. Same for your 12V power supply.
You can indeed power the Arduino with 12V if you connect it to the RAW pin. Not if you use the 5V pin (it WILL kill the Arduino). But you can't connect much to the Arduino or your regulator will overheat.