Persianized, Hellenized, and Carthaginianized Roman Republic

Although the Roman Republic's rise was pretty likely, it didn't necessarily have to happen in the same way that it did or with the same powers controlling the Mediterranean. If the cultural map of the Mediterranean had evolved differently, how is Rome in particular affected?

Persianized Rome:

Perhaps there was never an Alexander to defeat the Persians, or Alexander was decisively stopped and killed in Anatolia. Hellenism doesn't spread farther east than it already had, and the Achaemenid keeps a strong hold on their western territories. Later they have decayed enough to be conquered by Rome.

If Rome conquers a Persianized Egypt and Levant, how is that different from a Hellenized Egypt and Levant? Could Rome adopt some Persian aspects; if so, what would they be? Could Rome even adopt Zoroastrianism as an imperial cult later on?

Hellenized Rome:

Alternatively, in a different scenario, Alexander lives much longer, and Hellenization is wildly successful. A fusion between Near Eastern and Greek culture has occurred not just in the ruling class but among the population overall. The Romans strengthen Hellenization in the East as OTL, but they're so close to being fully Hellenized that they become cultural extensions of Greece. Or perhaps Greece would be seen as a less wealthy extension of Egypt and Mesopotamia.

In this scenario, where the population of Greeks vastly outnumbers the Romans, what happens?

Carthaginianized Rome:

Finally, Rome's biggest early rival, Carthage. If the Second Punic War had not been so devastating to Rome that they wanted revenge, or if the First Punic War was even more of a Roman victory to where Rome conquered Carthage and held the entirety of a foreign land--thus forcing the Romans to be less punitive and more syncretist--what would the effects of Punic culture have on the Republic?

Would the Carthaginians still be Latinized?
 
As far as Hellenized Rome, I'm not sure the Romans could get more Hellenized than IOTL. Any further and you run into cultural barriers. Chiefly, Hellenic philosophizing and Hellenistic science (scientific advancement, which was exploding in the Hellenistic era, stagnated under the Romans). The Romans prided themselves on being a martial culture of rough soldier farmers, and once Rome began to be "hellenized" you see these two contrasting views and lifestyles often clash in the late republic period.

Otherwise, Rome was fairly Hellenized anyway, its "republican" government system was taken from the oligarchies of if Greece and Etruria, and its organization of tribes is very similar to how Athens organized Attica.
 
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