This would be interesting. In my view, there would be three powers after the dust settles: Macedon, Persia, and Egypt. Of course, Persia could succeed in holding Egypt or Macedon could conquer it, but it seemed like Egypt was a fairly rebellious province, and with Macedonian control over Anatolia, the hold on Syria and Armenia is threatened. Each power would be under something of self-rule, with Persia ruling from Pars, Egypt ruling from the Nile, and Macedon ruling from Macedonia as powerbases, rather than having he situation of foreign conquerors ruling a conquered populace, which would have implications for state formation and stability as well as on the culture of these states. Hellenistic culture would also be far less widespread than IOTL. IOTL, Seleucid artifacts and cultural influences were found as far east as the Han Empire and Maurya Empire, and were large influences in Rome and Parthia. ITTL, Persia and Egypt block the way to he east for direct cultural diffusion by a ruling elite, though Hellene traders could always be a soft influence.
1. It was not an accident that Alexander could conquer the east. Macedon - an arguably much weaker power than Rome - was able to destroy them. I pretty sure that Rome can wrestle their mediterranean provinces away at the very least. The question is if Rome wants to do it.
2. Except the romans never warred with the persians. They warred with the parthians and the sassanians. Both Empire was based around the parthian nobility - the former fully and the latter mostly. They changed warfare in Persia. So does that still happen? Also the parthians and persians can be entangled in a pretty long conflict that Rome could use to its advantages.
4. That stalemate happened mostly around the OTL border Area. If they are fighting in Anatolia or Egypt the Iranians will have a much harder time than OTL: The things is that rome can ship his forces there from Italy. The core of the Persians was the iranian plateu - that is a much longer travel on land.
I agree it wont be an easy fight and Rome might not be willing to pay the price of those conquest unless he thinks its worth it. Without Alexandewr taking the gold of the persian Empire and dumping it on the economy the east will be much poorer compared to the incredible riches of OTL and that might mitigate the appetite of Rome to the point they stay away.
Also dont forget that the OP had only butterflied Alexander and he supposed that the macedonians were still able to wrestle control of at least Anatolia from the Persians. So stirring up the Ionic cities doesnt make sense.
And finally in the ancient world Rome was an absolute beast. It had an highly efficient and strong military and incredible manpower reserves - emphasis on the latter. See the 2nd punic war: which ancient power would have been able to keep fighting after that thrashing they got? The Persians were done in by 3 big battles OTL - and thats with Alexander taking the detour in Egypt and giving time to gather forces. As much bigger and more populous as Persia might be the ability of Rome to raise new armies was unrivaled for a very long time.
Rome will not necessarily rise, nor gain the hegemonic role it did ITTL. The Romans still have to deal with the Samnites, from whom they learned the manipular formations that served them well in defeating the more cumbersome phalanx. IOTL the Latin rebellion and 1st and 2nd Samnite wars were not walks in the park for Rome (Claudine forks, for example). Even if they beat the Samnites, they may have to face invasion from a Hellene successor state (Epirus might not invade ITTL, but who knows, if an adventurer usurps the Macedonian throne and decides to protect Megas Hellas, that may not bode well for Rome). Then the Romans may have to deal with Carthage for supremacy over the Tyrrhenian, Sicily, and the western Mediterranean, and the Carthaginians were pounding them pretty hard before they went down. Then Rome has to surmount internal issues, like restructuring its armies to accommodate a growing state. Rome has to somehow draw itself into eastern conflicts: remember that Rome actually initially had wanted little to do with the east, and it was only he combined fractiousness and disorder prevailing there combined with Macedonia’s inability to submit as a peaceful client that drew Romans into the east. Rome liked the tax revenue and plunder of course, but she did attempt to withdraw from Macedonia a few times ITTL before giving up and getting involved in the Hellenistic world. ITTL, with a stronger and more orderly Macedonia, who’s to say such a defeat and annexation would happen? Who’s to say that the Persian state would be in the exact situation the Seleucids were in, with decreasing control over the east, where their control was weaker, and a Parthian threat afflicting them at the same time leagues of Hellene states and Rome defeated and stalled them in the west? Who’s to say that a native-ruled Egypt won’t become confident enough in itself to fight off attempts to subdue or subjugate it? States never develop in a vacuum, and there is no guarantee that a state would follow the same course it did IOTL even with small initial changes. Rome will not rise because it did IOTL, it will rise due to a combination of internal and external factors as well as sheer chance, all of which cannot be taken for granted in any timeline, including our own.