Successful Rome's engineering and infrastructure

From what I read, both canals took some serious periodic maintenance because of silting, but the First Cataract Canal rather more so than the Canal of the Pharaohs. Nonetheless, in the latter case it was good enough for the Canal to operate on a scale ranging from several decades to a few centuries, since it seems the Canal was intermittently used from the Pharaohs' period to Byzantine times. Apparently it got in use when a strong state controlled the area and could deal with periodic maintenance, and fell in disrepair when the political situation did not allow to take care of the silting problem for too long. Since TTL an extremely successful Rome is assumed to control the Middle East since Classical times, it would certainly be able to keep the Suez Canal operative pretty much throughout the last two millennia, except perhaps during periods of temporary weakness, and when the empire recovers it would do the necessary fixes. In terms of economic and strategic value for Rome, the Suez Canal would have outmost importance, so I have no doubt a successful empire would undertake all the necessary maintenance and repair project, over and over again. Moreover, at some point Rome would build the modern version of the Canal, which cuts the link with the Nile and deals much better with the silting problem.

Well, the tech and knowledge for the Canal of the Pharaohs demonstrably existed even before Rome. As it concerns the modern version, I assume it could actually be done in premodern times if enough manpower and money gets thrown at the project (and TTL Rome would have plenty of both, controlling all of Europe, North Africa, and the Near East). The modern Suez Canal is fairly simple technology, a single-lane waterway dug across a desert plain with no locks system. Throw enough diggers at it, and it can be done, it doesn't actually require explosives or machines.  

For the projects that get built in Classical and Late Antiquity times, most certainly. It would be too early for a successful Rome to evolve beyond slavery. On the other hand, I'm fairly sure in a successful Rome certain OTL social institutions such as chattel slavery and second-class status of women cannot realistically survive all the way to modern times, stereotypes aside. Even during the Principate dynamics were noticeable in Roman society that suggest a gradual decline and marginalization of such increasingly obsolete institutions. Without racial prejudice to feed it and with ease of emancipation and assimilation of freedmen tp undermine it, Roman version of slavery would decline earlier and with less difficulty than the OTL early modern version, probably gradually transforming into a system of indentured servitude and apprenticeship. Of course, periods following major conquests when PoW would be plentiful might cause temporary relapses in the practice. But Roman society was and in all likelihood would stay drastically averse to the kind of ethnic polarization that would be necessary to entrench slavery for long heading into modernity.

My own notes for a successful Rome TL tend to assume gradual decline and marginalization of chattel slavery during the Late Antiquity and Middle Ages periods. On the other hand, temporary slavery as a form of punishment for convicted felons may well survive in Rome all the way to the 21st century, effectively making forced labor more or less the default punishment in Rome for serious crimes, when fines or community service would be too little and the death penalty too much. Much the same way, modern Rome may well have few moral scruples using enslaved felons for medical experimentation, in brothels, and the like. Another use of course might be gladiator games, which I expect may easily stay quite popular in some form all the way to modern times. Although I also expect over time the games would be increasingly performed by free professional athletes, again developing noticeable OTL trends, and include a mix of staged combats and extreme sports.

I greatly enjoyed the Romanitas series for its ideas, even if the author's prose made it somewhat difficult for me to read it in its entirety. If you ask my opinion, the series correctly nailed several features a successful Rome would likely develop, such as the multicontinental superstate extension, the bipolar or tripolar global dynamic with the other sophisticated Eurasian civilizations, and the evolution into an modern industrialized superpower not much different technologically from OTL developed world. If anything, it was somewhat too static (although perhaps the author made it for the sake of drama and familiarity) in assuming modern Rome would still feature chattel slavery as a major social problem, and stay an absolute monarchy with military coup as a frequent way of succession. A successful Rome would likely evolve beyond slavery (except quite possibly for criminal justice) and develop a more stable succession and check-and-balances system in order to endure. I expect modern Rome would have perhaps most likely develop a constitutional elective monarchy system not radically different from the US 'Imperial Presidency', albeit probably leaning more on the authoritarian side (e.g. lack of a notion of human rights for rebels and convicted criminals, successful and popular leaders staying in power as much as they like, the state getting exceedingly ample leeway to crush threats to its stability). So perhaps a bit more like modern Russia than America.

On the other hand, such a society would experience few or no ethnic/racial tensions. Roman culture's notion of civic identity and its universalist ideal were radically incompatible with racist prejudice, it had a definite notion of cultural superiority vs. less advanced/sophisticated civilizations but it expected anybody could be a good Roman with the right attitude and training. It would also have relatively few troubles with sexual orientation issues, since Roman society seemed to embrace the ideal of bisexuality as the default standard with most variations being a matter of individual taste, although overt effeminacy was and may well stay a major taboo given the social veneration of masculine values. For similar reasons, I expect Roman society would find a path to embrace women's equality, but likely an 'Amazonian' one based on its own values. In other words, Romans would come to internalize the notion women can and should be as tough, competent, stoic, and self-reliant as men, so they deserve equal legal and social treatment. Political correctness and reliance on 'feelings' and 'sensibilities' would likely stay alien and laughable to them even when women successfully claim an equal place at the table.  

Unlike most inventions that showed a tendency to independent rediscovery in different times and places IOTL and might theoretically replicate the pattern ITTL except Eurasian circulation of ideas often makes it unnecessary, blackpowder showed an important serendipity element, so it might go both ways but I lean towards discovery in a single place and a fairly rapid spread among the advanced civlizations of Eurasia, much like it happened IOTL. If anything, cultural and technological diffusion tends to be even more efficient ITTL across the Rome-India-China circuit, since the success of Rome first bypasses and marginalizes the Persian obstacle/middleman, then absorbs it altogether, seizes near-complete control of the sea trade routes to India, and prevents Islam from existing. Rome's imperial success is mirrored by China avoiding the isolationist, high-equilibrium stasis trap and India finding a path to imperial unity like its sister civilizations. Barring the inevitable differences for premodern civilizations, TTL Eurasia is much more globalized than OTL since Classical times.

As a consequence, the three civilizations share effective and lively trade exchanges, circulation of ideas, and diplomatic relations since Classical times. So cultural and technological innovations spread from one settled end of Eurasia to the other w/o excessive difficulty, Rome gets timely access to the Four Great Inventions (gunpowder, compass, paper, printing) and other Chinese technology much the same way China gets to tap Roman technological achievements (e.g. cement and engineering). This is one reason why Rome, China, and to a slightly lesser degree India influence each other into becoming millennial outstanding success stories as civilizations and empires all the way to modern superpower status, sometimes being competitors or rivals, sometimes being mirrors and analogues, but always being well aware of each other.  

Good and interesting questions. To address them I'd better check my TL draft and notes. Given my geopolitical leanings, successful Rome is one of my most preferred AH scenarioes ever and I've had a TL draft about it for some time. It covers in some detail Roman history in the first millennium CE (although it abstracts out all the dynastic stuff, which I'm terrible to grasp and write about) and the final state of the world in the Information Age. I've not yet finished it since I lacked time and focus to fully revise it according to evolving ideas and I have a writer's block about the missing stuff (non-Roman history, events in the second millennium). However I have notes and ideas.

I'm also uncertain whether the Roman and Chinese colonies in the New World remain bound to their empires for all time or an equivalent of the Atlantic Revolutions eventually occurs to create analogues of the USA, California, and Brazil. I also read another TL where the Roman Empire successfully pursues the 'Sahara Sea' project in Classical times and radically changes the landscape of northern Africa. I've been rather uncertain if the idea is truly realistic for premodern technology and whether to incorporate the idea in my own TL. Nagging doubt on this issue is part of the reasons why I made this thread.

Anyway, according to my TL draft and notes, Rome achieves its Renaissance stage of evolution in the 10th-11th century CE and engages in global exploration and colonization. Discovery of the Americas naturally occurs in this period through exploration of the Vinland route and the ever-ambitious Romans engage in extensive colonization of the new continent as soon as they notice its true nature and value. A repetition of Colombus' lucky blunder is neither necessary nor likely since Rome has all the access to sea trade routes with India and China it may ever need.

To be fair, at the time Rome is also more or less busy with the final conquest and pacification of rump Persia and Axum and first undergoing the colonization of Sarmatia in earnest, as its Gothic kingdoms suffer an analogue of the collapse of Kievan Rus. But I'm entirely confident this version of Rome has the resources and organization to wage colonization of Russia and the Americas at the same time. OTL Europe did colonize the Americas and half of Asia with an arm tied behind its back from its endless wars and the conflict with Islam. Here Roman Europe has its resources multiplied by greater extension, imperial unity, and Roman organization and skill, so it can do much more. It is also just as imperialist and greedy, although thankfully with no ethnic racism.

As it concerns premodern megaprojects, in my TL Rome renovates/builds the Suez Canal and achieves borders at the Vistula-Dniester line, Persian deserts, and outskirts of Ethiopian highlands in the 1st-2nd century CE. Settlement of conquered territories and extension of the Roman road network through them starts in this period and is completed in the following two centuries.

Once it fully recovers from the 3rd century crisis and engages in a fairly extensive set of domestic reforms to deal with the problems it revealed, Rome starts in the 4th century to build a canal network to link the major rivers of Western-Central Europe and the Med with the North Sea, which is completed to the Vistula-Dniester line in the following two centuries.

The Romans start building some fortifications on the new Eastern border once it is stabilized and it basically stands (barring a few minor adjustments and occasional breakouts of the steppe nomads) until Rome is ready to engage in colonization of Sarmatia. However they undergo construction of the real 'Great Limes' to rival the Great Wall after they repel a massive breakout of the steppe nomads (TTL equivalent of the Huns) in the 5th century. The crisis also drives them to annex the Bosporus kingdom and move the eastern border to the Neman-Western Bug-Southern Bug line, and the European canal network is of course expanded to reach the new border.

The Romans are intermittently at war with Sarmatian steppe nomads, the Gothic kingdoms gradually forming in Russia-Ukraine, Persia, Axum, and Norse raiders throughout the first millennium. These conflicts and the occasional bouts of serious domestic strife and disunion give them headaches and pause but they always overcome them and get the upper hand or fully recover in the end.

In the meanwhile Rome gradually evolves into a thriving proto-capitalist High Middle Ages/Islamic Golden Age civilization with no feudalism, marginalizes slavery, colonizes Arabia, and cuts down Persia and Axum from rivals to nuisances, preparing for their final conquest. In the meanwhile it also achieves religious peace and unity by developing a syncretic and philosophically sophisticated analogue of Hinduism as its dominant religion with an important Buddhist minority, and wiping out hostile and disruptive Christianity, Judaism, and Zoroastrism within its territories (remnants survive but are pushed in the depths of the steppes, Africa, and Asia).

It also develops a Senate elected by the empire's elites, a written constitution and succession law, a check-and-balances system and power-sharing balance between the Emperor, the Senate, the army, the civil service, and the landed trading elites. The foreign conflicts and the very extension of the empire inspire the Romans to develop truly ocean-worthy naval technology and an extensive optical telegraph system in the 8th-9th centuries to improve their logistic issues. However Roman naval technology was already good enough to allow extensive trade, cultural exchanges, and diplomatic contacts between Rome, the Indian kingdoms, and China since the beginning of the first millennium.

So Rome and China are entirely aware of each other as sister empires and civilizations since their first Principate and Han flourishing. They later become rival superpowers (with unifying India as the third competitor) after they both develop a global reach with early modern tech, but in this period they are distant and separate enough their relations are friendly.

Admittedly I did not give thought or explicit coverage to the other premodern megaprojects quoted in the OP when I wrote my TL. But my reasoned guess is the Romans do notice the value of the qanat system soon after they conquer Mesopotamia and western Persia, and gradually but eagerly spread it to all the other arid regions of their empire to maximize their value. It is pretty much the same way they devise and spread the heavy plough across northern Europe to fully exploit it soon after they conquer and pacify Germania and Dacia. As it concerns the Kiel canal in all likelihood they build it as part of the European canal system in mid first millennium, or at least as a reaction to the Norse problem in late first millennium. As it concerns the Corinth Canal, no good reason why they can't do it in the 1st century as they attempted to do IOTL, since the PoD is more or less in Caesar's or August's reign. I suppose explicit mention of these events is one more I should deal with in my eventual TL revision.