1) Yes you should be using the external ground because you should also be supplying the LED Wiz with 5v. This is because the LEDs you have connected could draw ~600-700mA on each LED Wiz and you only have ~500mA available from USB.
You need to feed the LED Wiz 5v and use the ground terminal to connect to the power source ground.
Ok, this is striking me as a bit off for some reason. You don't need to connect 5v to the LED-Wiz board anywhere but the "Bank Voltage Select" terminals. If the wire heading out of the photo is going to the PC power supply, it's not correct. That terminal is a tap into the 5v USB power, and shouldn't be tied to the PC supply.
But if that wire is going to your LEDs, that's fine, as long as the load is not greater than the 500ma spec of the USB port. This is what I thought was happening.
If you are going to the PC supply for the +5v, you don't want to hook anything to the "USB +5v" terminal at all.
RandyT