OK, a number of things from the previous discussion.
First and most important - safety!
I see that your power cord has a green wire. I presume it has a three pin plug. Having removed the plug from the wall socket, you need to start again with the terminations by cutting the black and white back to the same length as the green, then remove the sheath for about 2½ inches so all three wires are available, strip the ends ¼ inch, twist the strands together so they stay together and "tin" just the end with solder. Then connect the live and neutral to the first two screws and the ground (green or green /yellow) to the third which is marked with the "earth" symbol.
Now you have a safe power supply (as long as you do not touch the terminals! ).
You may have been confused by the general reference to "ground" or "GND" in Arduino projects. This refers to the negative reference, but is not always going to be actual ground - it may or may not be connected. The power supply however deliberately does not connect the negative to ground but its "earth" is the connection to the metal case and you do want that to be connected to your mains ground to prevent it picking up any spurious voltage from its internal parts. In fact, should a part somehow short out to the case, you want that to blow a fuse instead of making the case "live".
If two of them are for voltage sensing, not having them connected is going to become a problem.
No, they will not be for voltage sensing in this case (not that heavy a current supply), but for parallel connection to distribute the quite heavy current rating of the power supply. Clearly the terminals are in order: "L", "N", "earth", "-", "-", "+", "+".
With only a few inches between power supply Arduino and LED strip, you would not need the 1 mF capacitor directly across 5 V and ground at the LED strip, but with longer leads you do. Since the Arduino is (presently) separately powered, you should have a 470 Ohm resistor in series with the data line at the LED strip.