Welcome back. I remember those long sun up to sun down days back in 1950's.
I'll review Dan's Plaque # 8, wonder if it still works ?

 

I'm not sure what your question about shift registers is so here is an expanded drawing of 74HC164. See that it is composed of a string of D FF's. The data on the D input will be shifted into the FF on next clock. In our case, the first D is connected to ground so that with each clock the string of 1s advances until the first SR is full. As its output is connected to the next stage D, the next stage is filled with 1s until the last stage of last SR it filled.
When last stage goes high, it creates a reset pulse, clearing all stages.

 

Just a small change in clear. Added R56 & moved C1. R55 is pull-up for open collector U6-16.
you have a message as noted by the tic mark by your avatar.

 

Do we have a plaque size & LED color yet ? My guess- 16 cm dia. , green ?

 

Do we have a plaque size & LED color yet ? My guess- 16 cm dia. , green ?

Probably 0603 green LED SMDs

 

If a reset on power-up is desired, then for DP 8A a high would be applied to pin 4 U6 & pin 15 connected to 16.
If not resetting every thing would be normal after the first cycle. For 555 clock, see post # 114, U1.

 

Also, If I have two LEDs lit up in parallel, I usually just use individual resistors but I'm aware that in this case I could very easy tie the cathodes together and use just one resistor (though it wouldn't be as bright)... In this instance, do you think It would be wise to do that? I'm already thinking I'll have to use two different values (one value for when two LEDs go high; and then another value when 5 LEDs go high, just so the current flow stays even) as I'm trying to keep the same brightness.

 

Here is an example: 5, V matched aqua LEDs in parallel, 47 ohm current limiting resistor, one stage of ULN2003A, 5.21 V battery. V drops, ULN - .753 V, LEDs- 2.66 V , R- 2.55 V. From R drop, I= 38 mA, or each LED = 7.6 mA & quite bright. As ULN drop is current sensitive , R s
might have to be adjusted to give desired LED current . One R per LED or one per group of matched LED has to be your decision, my opinion, either will work.
Correction, used wrong value, LED current = 10.8mA, total 54.2.