If so, when could one say that China experienced a 'fall of Rome' moment?

Yes, sort of. The collapse of eastern Han. General disunity (punctuated by a brief Jin restoration) pervaded until the Sui dynasty some 350 years later. The interregnum was characterized by things like short lived and fractured regimes, the spread of Buddhism, and large scale population migrations. Tang China would look quite different from Han China. Paging /u/lukeweiss or /u/jasfss or somebody else, the details of this era are out of my depth. Overall this whole period doesn't get nearly as much attention as it should imo.

As far as the 5000 year claim goes I wish we could tell you more, I don't know if we have a Chinese archaeology expert here who could elaborate on the very distant past and the meaning of sites like erlitou or taosi. I will say though that the 5000 year claim projects some anachronistic ideas back into the past. For example Sima Qian draws a continuous history from idealized days of myth through the Shang and Zhou kings, but in reality you can identify a number of discontinuities in early Chinese history if you look hard enough. I don't know if it's possible to identify a precise moment in time when the idea of a continuous civilization at the center of tianxia (中国) became a widespread. Await answers from ancient / early imperial China experts. Perhaps /u/keyilan can trace the usage of 中国 for you.

hi! not discouraging more discussion here, but there have been a few posts on this topic; check these out

• When did the view of "China" as a continuous and singular country/civilization emerge? - includes a link to one more

• Please give this foreginer living in China some 'outside' world opinion: is it true that China has the longest continuous civilization? but... and some questions on Mexico - includes links to several more

if you have followup questions to locked threads, ask them here & include the user's username so they'll be auto-notified