For some reason, the game almost always starts asking me about Syncing Saves. Should I do it from the Cloud or from my computer? Why and how did this happen? Of course, it would be great to fix this.

Another problem is that Syncing Saves often returns an error, something like "The Server is Overloaded and the Sync Was Not Correct" (or something like that). Of course, it would be great to fix this and get everything working properly.

Are you actually playing cross-platform, or only on your PC? Cloud save sync is pointless if you're only playing on a single device.

First thing I recommend is to try launching the game directly from the main .exe, located in:
[X]:\Cyberpunk 2077\bin\x64\Cyberpunk2077.exe

This will bypass GoG Galaxy and should not prompt for any cloud saves. You will see a cloud sync error in the main menu, confirming the feature is disabled. At that point, saves are saved and stored locally on your PC.