matledoublev:
I'm a bit confused regarding this part. The red (power) and blue (ground) horizontal lines connect the LEDs, do they not? Also, why would I have to feed power and ground to the left side as well?
Yes, they are connecting power to the LEDs at one end of each strip but as has been explained, the foil in the strip is not very substantial and you tend to lose voltage by the other end of the strip.
So you want power to connect to the other end of each strip - the top end in your diagram, so you need them all connected together just as the bottom, and power fed to that common connection as well. Even if you did not take the power to the top common, there would be some beneficial cross-feed from strips which were not illuminated and thus not taking power.
matledoublev:
Another wiring arrangement I've considered is to have all of the data directions aligned, I would just need longer wires to run from the top of the LED column down to the bottom of the next LED column. But based off my knowledge this would be suboptimal, since a longer wire would mean more voltage drop, if I'm not mistaken.
I do believe the NeoPixel library provides support for the "zig-zag" arrangement in an array.
It is not that you lose voltage in long data lines between strips, but you risk picking up interference because the PWM operation induces currents in the ground and power lines along the strips so the voltage at one end is not the same as the other. Best to have the data and ground connected by the shortest possible path.
matledoublev:
I read somewhere that it would be preferable to use an external power supply and not the Arduino Uno 5V supply, since the large current draw from the LEDs exceed the UNO's specifications. Another poster mentioned that I must have a common ground, so I was thinking to setup the wiring connections like this:
The current for the NeoPixels should never pass through the Arduino. The only time it might would be if you powered just a few (perhaps five) NeoPixels from the "5V" pin when power is supplied through the USB connection.
But you should power the Arduino from your regulated 5 V supply - which you already have for the NeoPixels, so you take that back from where the data connects to them, to the Arduino together with the data. This carries only the current required by the Arduino.
You might however wish to put a 1 Amp fuse on this 5 V line just in case you accidentally short it on the Arduino which might otherwise go up in smoke if that happened. 
Did someone mention? A UNO/ Nano will only control about 300 NeoPixels. 