1) When I engage the brakes, the rear lower right tail light glows "low" (only one filament) while the left glows "high" as designed (both filaments).

2) When I turn on the city/parking lights, both sides glow "low" as designed but when I engaged the brakes, the left side glows "high" as designed but the right side doesn't not light up at all.

I checked all the fuses and they are fine. Any suggestions where to start troubleshooting?

 

Have you tried replacing the bulb? You could possibly have the wrong bulb installed, I've seen this happen before with other cars...

 

I tested the bulb with my 12v power supply and both was able to test both filaments without issue. I also swapped the bulb with the left side which works fine with the same result.

I think the problem is within the bulb socket...