A seven volt 10 amp power source has been acquired and should provide enough current
to drive the Arduino and the 256 LEDS but not sure how to connect
the power supply to the project.
The problem is that it will provide the current the voltage is too high to power the LEDs. The Max7219 requires 5V, so you have to regulate your 7V to that. Regulating an amp or so is but the 5A you need for your LEDs might be a bit more tricky. I would suggest either you get a 5V power supply or a couple of buck ( switching regulators ).
You can power your Arduino on the 5V pin and likewise for the sensor.
Then all 5V and ground lines can be shared. If you have a separate power supply then all grounds must be connected together only.