AWG 12 would be my choice while AWG 14 would likely do fine but considering the cost difference and a short run I would still go with AWG 12. Since your uC is alone I would just use a buck converter to power it off your 12 VDC supply using maybe a decoupling capacitor at your uC board.
What do you mean by fusing? Do you mean using a physical fuse? Or did you mean fusing multiple power lines together and taking off smaller amperage per line?
The barrel jack that I have surely isn't rated for 15amps, so I'm shopping for a new barrel jack that will handle this kind of power and mount to a prototype board, but I'm coming up short.
Does anyone know where I can find a barrel jack that can handle this amperage?
Hard to size the wire as the distances are not known. Also the plug strip will not support current over 1A, sometimes they are less. The current limit of the power supply should protect your wiring. You can use a wire calculator to determine what size wire you need. http://wiresizecalculator.net/ Here is another calculator: Voltage Drop Calculator | Southwire You will be getting some surprisingly heavy gauge sizes recommended. As previously suggested run multiple feeds to the led strips, one feed every foot. In your case it appears end feeds would be the best. You can also set your power supply for about 5.5 Volts to compensate for the voltage drop in the feed wire. Higher voltage LED strips would cut wire gauge and in place of the buck converter use a SEPIC converter as the voltage can go below 5V. This should get you started.
I use a Nano or tiny88 on newpixel 3m (16') WS2812, 60pix/m with 12v 5A laptop power supply, running marquees 24/day.
Electrolytic capacitor between power and ground at the strip.
Resistor on signal pin between strip and Arduino.
No power injection, and no color loss (to my eyes).
Power jacks are "barrel" (amazon) center F / ring M, like laptop chargers.
The code never uses 255, 255, 255 (RGB). Most of the time 127, 127, 0 (very bright at night).
Something to consider is the heat generated by the LEDs. It looks like you sign has a front (plexiglass) and a back (unknown). All 60 watts will be trapped within your front and rear.
