Why did much of the Western Roman Empire adopt Latin, while the Eastern Roman Empire remained Greek speaking even after Roman conquest?

So before the Roman Empire included the east, Greek was the “lingua franca” of the Eastern Mediterranean and Asia Minor, and because the Roman elite respected Greek as a philosophical and literary language, they left it alone in the East and indeed made it one of the two official languages of the Empire. So for a Greek-speaking resident of the East, you could still continue your day-to-day life and even interact with the Roman state, all without having to learn Latin.

The same factors weren’t really there in the west, where there were a lot of different languages, none of which had the prestige and thus utility of Greek. And while the Roman Empire didn’t try to impose Latin on conquered people in the West, because Latin was the only language of administration, law, military, and business, there was an incredibly strong incentive for the children of the provincal elite to be educated in Latin so that they could become Romanized and advance in society.

That being said, it should be noted that bilingualism in elite circles was the rule in both the West and East through the 5th Century CE. Indeed, for a long time, because Latin was considered the “language of power,” there were complaints that the study of Latin by high-status residents of the Eastern Empire was harming the quality of traditional education in Greek rhetoric.

What was the effect of the Roman Empire being able to tax enough to support its army upon its populace? With medieval governments being smaller than the Roman one, did that mean a peasant in medieval Europe was taxed more lightly and was economically better off than his Roman counterparts? Or, was the Empire able to tax the wealthy more effectively, or have a more productive economy due to the sort-of globalization inside the Roman world? Or, did medieval taxes just support nobility lifestyles?

The main effect seems to have been a great deal of internal economic growth and peace, as we can see from the fact that Roman cities grew in size but didn’t build walls until the crisis of the third century.

In terms of your second question, I lean more towards the effective taxation of the wealthy angle (at least at the height of the Empire’s power), as medieval taxes tended to have higher rates and more regressive forms to make up for the fact that the medieval state often didn’t tax the wealthy effectively.

Calculating GDP per capita in premodern eras is extremely difficult, but it does seem as if there was decline or stagnation at best in living standards between 1 AD and 1000 AD, with a slow recovery during the High and Late Middle Ages, which picked up steam during the Renaissance and Early Modern eras, and then really took off following the Industrial Revolution.

Why did the hunnic invasion lead to the Germanic people migrating west but the rise of Rome didn’t cause a eastern migration

I’m guessing that by the rise of Rome, you mean the expansion of Rome north and east into Gaul, Germania, Illyricum, Dalmatia, Pannonia, Dacia, etc.

I think part of the reason is that the Romans actually didn’t push that far northeast; the decision to stop at the Rhine and the Danube limited the “push” factor eastward. Likewise, the Roman policy of trade and tribute, establishing client kingdoms in Germania, Dacia, Sarmatia, etc. created a buffer zone that could absorb eastward migration.

And sadly, I think another part of it has to do with the Roman practice of imperialism, that the focus was on conquering and then Romanizing people where they were as opposed to driving them off the land, but even more so the focus was on enslaving people, which meant that people who might have become refugees to the east instead moved south into the Mediterranean slave markets.