I have assembled my Bulbdial clock (monochrome) up to the point of attaching the Red ring. Prior to this point, the Blue and Green rings worked perfectly as I went through the alignment and testing steps. After adding the Red ring, problems developed.

All LEDs on all rings light when they are supposed to. However, some LEDs on the Green and blue ring also light when they are not supposed to. The improperly lit LEDs can have brightness levels of full bright, dim, flickering, or come on then fade out. The pattern is always the same - selecting one specific LED always causes the same set of improper LEDs to light at the same brightness level. (e.g. selecting Red LED 7 always also lights Green LED 13 at full brightness, Selecting Blue 3 always also lights Green 1 full and Green 3 dim. Note: I am counting clockwise with the LED at the 12:00 position as zero.)

This does not happen right away. If the unit is left unpowered for a long time (like overnight) the unit behaves normally for a couple of minutes.

A visual inspection reveals no obvious shorts or solder bridges.

To me, this looks like what the Wikipedia article on Charlieplexing describes as the effect of a bad component (a leaky or internally shorted LED).

where do I start identifying the source of the problem in my clock?

I have seen behavior like this in a clock that had two bad solder joints from the main board to the jumpers, so there isn't necessarily a bad component.

The difference between cool and warm components sounds like a flaky solder joint problem. The first thing that I'd suggest is to resolder all ten of the connections from the main board to the jumpers (LED1-LED10). Make sure and add fresh solder to each location, and that they end up shiny and wet when you finish. Then do the same for the other ends of the jumpers, at the red ring. If you can reach the locations at the green ring, you can touch those up as well.

Next, check every one of your LEDs and make sure that they are all facing the correct direction. Each one has a flattened side of the "collar," on the side that faces towards the rounded solder pad. (A reversed LED is indeed a "leaky" LED.)

Then, go through in alignment mode, and double check to see if every LED is lighting up when it's supposed to. If you find one that doesn't, that can be a big clue. It is more likely that this would be due to a bad solder joint somewhere (not even necessarily at that LED) than due to a bad LED.

If things aren't working by the time you finish these steps, then the next step will be to carefully document which LEDs are going on and when-- by looking at the circuit diagram, it should be possible to tell which LEDs and/or connections to suspect, and I can help with that stage.

That issue has mostly gone away.

I verified that all LEDs were in the correct orientation. I tested nearly every connection with my multimeter and found no discontinuities. I tested all close-together solder pads and found no bridges. I tested all inter-ring zerohm jumpers against each other and found no cross connections. I then re-worked all of the joints on the inter-ring jumpers, and a few other connections that were less pretty than the others. I scrubbed the areas around each solder joint on both sides of each board with solvent and a toothbrush to eliminate any possible skin oils, solder droplets, flux residue, or other debris.

The clock now operates normally, with incorrect LEDs lighting only occasionally and then only extremely faintly (I have to be looking directly into the lens of an improper LED to see that it is lighting at all). This is only detectable at the highest brightness setting, the improper LEDs are not visible from a normal viewing location, and they do not cast shadows on the clock face. I consider this problem solved.

The clock now operates normally, with incorrect LEDs lighting only occasionally and then only extremely faintly (I have to be looking directly into the lens of an improper LED to see that it is lighting at all). This is only detectable at the highest brightness setting, the improper LEDs are not visible from a normal viewing location, and they do not cast shadows on the clock face. I consider this problem solved.

Yes, that sounds quite normal indeed.