The HRE for much of the time between 1190-1260 was ruled by the Hohenstaufens, who also controlled the Kingdom of Sicily after Henry VI's reign. Henry VI notably attempted to make the crown of the HRE hereditary, and his son, Frederick II, was noted as much favoring Sicily where he grew up and was almost uninterested in Germany by comparison. However these events drew the enmity of the Popes, who found the Papal States surrounded to the north and south by the Hohenstaufen, and subsequently did all they could to break up that personal union between the Kingdom of Sicily and Hohenstaufen Holy Roman Emperor. If Henry succeeded in making the HRE a hereditary monarchy and Frederick managed to clearly triumph over Innocent IV, the emergence of a Italy centered HRE isn't impossible. Whether this would be at all a successor to the Western Roman Empire is debatable, but the Hohenstaufen were Kings of Jerusalem at the time and the Latin Empire would fall in 1259 only a decade after Frederick II died. A still strong Hohenstaufen could form an alliance with the new Byzantine Empire for recognition as the successor Western Roman Empire. A 'renewed' Western and Eastern Roman Empires could potentially cause a cultural renaissance for those with Greek and/or Latin cultures.
As for China. I'm far from an expert, but I'd say it would be difficult because of geographical reasons. The Mediterranean connects so many places it is a rich trade opportunity. This gives everyone around the Mediterranean a good reason to want to break into that market by gaining a reliable port. Rendering it a Roman Lake forged a situation where there'd almost always be multiple opponents waiting for a moment of weakness to take advantage. Eventually you have a dozen different countries taking chunks of the coast to end up like OTL. By contrast Chinese civilization developed along the Yellow River, which is not as large and unmanageable a target. The Grand Canal then extended the 'cradle' of Chinese civilization onto the Yangtze River, a valuable means of transporting goods over a large distance while also requiring a powerful government to maintain. A fertile, developed, and yet still relatively small region thus allowed the Chinese to support a massive population and subsequently spread outwards.
Simply put, sheer size makes clear any civilization holding all or most of the Mediterranean is going to be forced to deal with constant invasions and outwards forces pressuring them. It would take an organized, prosperous superpower greater than all around it to gain, which the Roman Empire was, and even the Roman Empire had times of strife and civil war. This meant holding it more than a few centuries, which it did, becomes increasingly unlikely. The best you could hope for was a Byzantine type successor kingdom for the 'Romans'. Having a second Roman Empire subsequently conquer the Mediterranean becomes increasingly unlikely as time passes, the Barbarian kingdoms build on the ruins of the Romans, and multiple cultures develop along the Mediterranean. By contrast the Yellow River, Grand Canal, and Yangtze River form an effective base to work from. While it could be split between multiple kingdoms, like during the Warring States Period, they're close enough and naturally are interconnected enough that there's a definite push to unite this area into one political unit. Once a state controls this cradle, they have the population, industrial, and resource capacity to start expanding outwards into the areas that would become traditionally dominated by China. Outside forces could conquer the Chinese, but that dense population base would almost always outnumber their conquerors enough that they assimilated to the native Chinese. To stop this, you'd need to permanently fracture the Yellow-Yangtze cradle. I'd say once the Grand Canal is built by the Sui, the trend towards a powerful central state is pretty set. Not completely so, but you'd need something major like the Mongols maybe deciding to destroy instead of conquer, destroying the Grand Canal and basically destroying the Chinese civilizations through outright genocide.
So the real problem is the Chinese civilizations are pretty set from a geographical point, an almost natural occurrence of a long lasting, great civilization. The Roman Empire took near half a century to reach the borders of its zenith, and was a consequence of its military, logistic, and organizational capabilities compared to its neighbors. Once these started to break down, the Roman Empire declined and was ultimately torn apart. The Chinese dynasties would fall, but naturally gravitated towards a certain base that simply put would always make the Chinese a major power.
Having a successor to the Roman Empire is entirely possible, and actually occurred OTL to some degree for over a millennia after the Collapse of the Western Roman Empire. Having a Roman Empire hold the territories of its zenith is near impossible, requiring the Romans to have advantages over all its neighbors in multiple areas despite centuries or millennia of interaction. A divided China is almost just as difficult to achieve, either needing a POD early in the formation of the Chinese identity or a major catastrophe on the level of the treatment the Mongols gave to a number of civilizations. Anything else is likely to end up like the half a dozen other times a major Chinese dynasty fell, only to eventually have the successor largely rebuild the exact same base back up again.