There are some great thread on what to stock to prevent so many waits. @LarryD is very helpful in organizing that info.
I’m still waiting on shipping for a few things, but this capacitance of the wire thing rang true for me. I redid everything with 22awg stranded wire, which I had on hand and which is the size the original wiring). I had been using 20awg. This actually helped a lot. Now all six LED strips (2 original and 4 new ones) are acting properly except one of the original strips is flickering a bit. It’s following the data line and doing what it’s supposed to do! But it’s flickering a little. Big improvement.
I don’t really understand how to use a 74HCT04 in this setup. I know this is a dumb question for people who understand these things much better than do I, but is there anything else that I could use to condition the data lines to keep their shape properly? Something that can be put in line with the wiring to help me keep this simple and (relatively) easy?
Using a 74xx04 is not very complex, you can simply do this
If you look at the pinout, you will see that it simply has 6 Not-gates and a VCC and GND pin. You should power it with 5v, and though the arduino's 5v is probably enough to power it, i recommend you power it from the 5v power supply.
Have I missed something here?
I can't find any description of what "Arduino" as actually being used here. The use of a 74HCT14 or 74HCT04 is only relevant if using a 3.3 V logic microcontroller, otherwise a 74HC14 for 5 V logic.
The Arduino should be powered by the same 5 V power supply as the LED strips. The power to the Arduino should come back to it along with the data wire from the common distribution point - which is where you would mount the 74HC14. The series resistors should connect at the "DIN" of each strip, not at the distribution point.
If using a UNO, (only) the "5V" pin should be disconnected from your 5 V main power supply whenever you connect the USB to a PC for programming.
Please read again my reply #13
Hmm. OK, I did read that - again. 
Is there supposed to be a point in your comment #25?
That is just not true ! How clear do you want it explained ? The use of TTL chips in a 5v logic system is totally relevant if you use them to split the signal and by doing so, reduce the total capacitance that the signal has to deal with, and in that way manage to keep the timing of the square wave within tolerance.
Really i have seriously overexplained it now, if you still don't get it, i give up.
28 AWG is even better, though getting a bit fragile.
You can also use solid cable of a thin gauge if the installation doesn't move.
Less surface is less capacitance.
If you are not using a device which operates at TTL logic levels (0 V for LOW, 2.5 V for HIGH) then there is no point using a chip whose function is to convert from those logic levels.
Perhaps more to the point, if you are actually using this as a buffer some distance away (over cabling) from the data source, being a 5 V Arduino (UNO, Nano etc.) then choosing a device with an asymmetric threshold (1.25 V instead of 2.5 V) will actually distort the timing to an extent dependent on the capacitance of that first run of cable. 
I cannot see why you would deliberately want to do that?
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