Thanks for your response, Retiredcaps! Much appreciated.
However, I think I might not be able to solve this one. Here is my follow up:
5. As I don't own an Oscilloscope (yet), I don't know how else I could measure a clock signal. Using my cheapie multimeter I switched it to 2000mA range and was able to measure that the voltage across the oscillator (Y1) is around 20mV. But not steady. I assume this is due to the fact that the measuring multimeter is trying to determine an average voltage over time, and the voltage is permanently varying. I guess with the limited measurement tools I have, the only conclusion I can draw is, that the oscillator is probably working fine. But that was a good trace, never the less.
6. You are right: I haven't noticed that before: RX2 is also unpopulated. And LED1 and LED2 as well. Hmmm. Seems to be nothing out of the norm then...
7. Thanks for the picture quality compliment! Took me quite some effort to build up the tripod and all camera gear for those shots. Yes, indeed the components you mention seem to be hand soldered in. I have tried re-soldering Rx1 using my brand new heat gun. It looks better now. But it didn't fix the multimeter.
8. Q8 looked almost like it had an edge broken off on the picture, indeed. But I have checked it again, and it was indeed just some flux residue. After cleaning it up, it looks totally fine.
I should add that yesterday I was sometimes able to get a faint reading on the display of the multimeter when switching through the different modes. But the reading was weird and didn't make any sense. And it always faded after a second or two. And today I am not able to produce the faint readings on the display anymore. Even yesterday I was initially not able to produce these faint readings. I seemed like a random act of kindness of the multimeter. Like throwing me a cookie.