Successful Rome's engineering and infrastructure

Theoretically speaking, a Roman Empire that avoids its OTL decline and absorbs the rest of Europe and the Middle East might still suffer a lasting West-East split similar to the OTL one, due to factors like the Latin-Greek cultural division and the Roman space's tendency to split in western and eastern halves in times of crisis. Anything more than that would be impossible if the decline spiral is avoided and the Germanic, Slavic, Persian, and Arab enemies are assimilated and turned into resources before they can become a threat. But it would likely require something like lasting religious antagonism to make it a likely occurrence and get it entrenched. As a matter of fact, such a dual division is much more likely in a scenario where both the Carolingian and the Byzantine empirs become success stories than as an evolution of Classical Rome that overcome the domestic and external causes of its decline. Otherwise, the universalist character and organizational efficiency of the Roman civilization suggest an evolution towards a Chinese dynastic cycle trajectory, where all strife and division periods are ultimately temporary.

As a matter of fact, in my TL a couple such periods are factored in during the first millennium, but they last a few decades to a century at most, and are universally seen as aberrations. I'm tentatively assuming the dynastic cycle pattern is still present but much less frequent and damaging for all TTL civilization-empires for various reasons (more efficient cultural progress thanks to greater circulation of ideas and mutual competition/imitation, greater resources and less threatening external threats from more successful expansion). So my bet is the dynastic cycle tends to lessen more and more the older and more successful Eurasian civilizations become, to be almost entirely erased once they achieve Early Modernity or so. E.g. I expect the Mongol rampage to be much less damaging for Eurasian sedentary civilizations than OTL, more like a most serious border conflict with some painful damage in peripheral regions than a takeover or devastation of Europe, China, the Middle East, or India. Of course, major convulsions such as an analogue of the Atlantic Revolutions are still possible and even likely at the onset of modernity, but likely not so much as to cause a collapse or breakup. A possible exception might be a lasting independence of faraway colonies such as the New World ones.

Yep. Far from inevitable or especially likely, and perhaps not the most likely outcome, but also far from impossible in the right circumstances. Uncertainty about this is one of the reasons that prevented me from completing my TL; the other issue is a writer's block about getting the same level of detail for the non-Roman events. Anyway, I'm trying to revise what I have available according to new ideas ad make it fit for posting. It is more or less the only other exception I'm willing to acknowledge as plausible to TTL Rome's effective immunity to permanent breakup. Because of distance, it looks a lot more feasible than a lasting West-East split, which would likely require something like the two halves embracing different and antagonist religions or denominations. Even so, it would require something like a neo-republican and democratic revolutionary ideology getting momentum in the Roman colonies in the New World and China giving support to the rebel colonies to weaken its rival. For that matter, the same scenario is more or less equally likely to happen for the Chinese colonies in the New World.

I can try and include this outcome in the 'final snapshot' segment of my incomplete TL, where it makes a summary description of the world outside modern Rome's borders, if and when I am done revising it for posting (I have yet to edit the last century of the first millennium, when Rome transitions into Renaissance, and a few bits here and there). Unfortunately I have a few general ideas about what may happen in the second millennium, but my writer's block prevents me from writing a real TL description about it in the foreseeable future, much as I love this scenario. However I was able to write the final snapshot portion, originally as an ISOT.

Kind of. The Imperial executive system TTL Rome evolves combines features of constitutionalism (written constitution and separation of powers), elective monarchy (the Emperor normally rules for life and the designated successor needs approval by the Senate, or by plebiscite in modern times; the Senate and in modern times the Cabinet elect the new Emperor if the established line of succession breaks down), presidentalism (the Emperor has vast executive powers and is the most influential and prestigious component of government, but the Senate remains very important in its own sphere), and a succession system that ITTL replaces hereditary monarchy, appointment by the incumbent.

It has all the conceivable pomp and charisma of a powerful Imperial monarchy that rules over half of the known world. The Romans like stability and efficiency, so they do not mind keeping a sufficiently competent and popular leader in power for life, nor they have any ideological objections to nepotism if it works. On the other hand, they remain sufficiently anti-monarchical and practical they never accept to make automatic heredity the main succession system, nor for that matter they ever create a blood aristocracy. Their society is certainly quite class-based, hierarchical, and based on patronage networks, but is pragmatic enough it never loses sight of the positive value and importance of meritocracy, so social mobility always remains quite feasible for the sufficiently talented. In this sense their elites are rather more similar to modern capitalist ones than OTL European nobility.

Therefore they develop their political system as a compromise between the strengths and flaws of elective and hereditary leadership. If the incumbent Emperor happens to have a relative or protege that looks sufficiently competent and popular, and proved his worth in public life, his succession gets accepted w/o question. If the supposed heir is lacking in the Emperor's family or close circle or the potential choices look like a Caligula or Charles II of Spain case, he gets sidelined and a more suitable successor gets picked among the Roman elites at large. They long ago picked adoption as a widespread tool to settle succession problems when genetics or fertility prove subpar, so they don't really have any blood purity concerns. For the Romans, the poor talented commoner boy that joins the army or the civil service, becomes a successful general/administrator and statesman, and impresses the Emperor and the Senate into making him the successor to the Imperial throne is a rags to riches tale that proves true in reality multiple times in Rome's history.

As it concerns the Imperial cult, when the Romans successfully reform their state religion in sophisticated paganism, it transforms into veneration of the personified Roman state rather than divinization of the Emperor. There is a sense Rome as an empire and a civilization got a divine mandate by fate to bring peace, order, prosperity, and good government to mankind. The Emperor is the apex and face of the Roman state and the most recognizable avatar of this mandate and so he commands exceptional authority and respect but he is not seen as divine or infallible, nor his bloodline as any more special than a modern political dynasty. The Principate vision of the Imperial office more or less remains true with the few tweaks described above throughout TTL Rome's history, with a more successful trajectory the switch into the Dominate model never happens. So the Emperor remains the exceedingly powerful and influential, quasi-dictatorial leader for life of a republic. It never adopts the personal excesses of elaborate choreography, etiquette, or veneration seen in other OTL monarchies. Then again, the Romans do love to ostentate their power, wealth, and successes a lot in everything from architecture to spectacles to public ceremonies, and this certainly reflects on the leader of their state.  

I'm not really sure what you mean here.

Since IOTL the Western political systems had a wholly different trajectory after the Dominate and the fall of Rome, it is not so easy to find a simple analogy among them for the system TTL Rome develops. It has a constitution and a certain degree of separation of powers, but it is not exactly a liberal democracy. It has a powerful executive monarchy, but it is neither precisely hereditary or elective, although it has features of both. It resembles a monarchy, but it has no true nobility of blood or divine right of kings or automatic heredity. It uses a succession system (appointment by the incumbent ratified by electoral vote) that was relatively rare IOTL but becomes the norm here. Certain features do seem more akin to a semi-dictatorial presidency-for-life than anything else.

Sorry, a question.
"Bronze swords" - engineering products of the Roman Empire?