Roman gods were modelled after Greek gods, with major gods having one-to-one correspondence in terms of referring to the same entity (although I'm sure there were slight cultural variations in the attributes and myths/legends of said entity, or at least the ones considered most important).
Did the Greeks agree to this interpretation, both before and after Greece was conquered by Rome? As the original "authors" of their Pantheon, and generally pretty snobbish towards any non-Greek cultures, were they proud or offended at their pantheon being renamed, and possibly mutated, by a foreign culture (which ended up dominating over their own)? Did they just see the Roman pantheon as an uncontroversial translation difference, or did they see it as a second-hand copy, derivative, cultural appropriation, or even heresy? Would a Greek visiting Rome pray at a Roman temple to Jupiter exactly as they would pray to Zeus back home?
A related (and broader) question would be asking about the Greek world had a concept of heresy, or taking offense when other cultures (or factions in their own culture) had a different interpretation of the nature or role of the gods, priests, and religious rituals, compared to the "mainstream". I know the Greeks had a much less personal and dogmatic relation with their gods than the Judeo-Christian tradition, so I wonder whether differences between pantheons such as the Greek and Roman ones would be along religious dogma lines, or rather simply along cultural/tradition lines. Were religious "schisms" a thing in the pre-Christian Greco-Roman world? Did they ever escalate to a major political issue rather than a purely religious one?
You may find some food for thought in this answer that I posted to a related question a couple of weeks ago. The most salient points there are:
Mars is the Latin word for Ἄρης in the same way that ambulo is the Latin word for βαίνω.
Even in the Greek and Roman pantheons, not all mappings were one-to-one. When for example the names of the planets got turned into Latin, the Greeks could refer to the planet Mercury as either 'the star of Hermes' or 'the star of Heracles', but the Romans used only one name, Mercury.
The mapping of Roman gods onto Greek gods is mostly a product of Roman literary authors telling stories about the Greek gods in Latin. That process is relatively one-sided: Greek literary figures didn't tell stories about Roman gods so much.
But where Greek authors do talk about Roman gods, the answer is yes, they did translate names into Greek: the earliest example is Faunus becoming Agrios in the epilogue of the Hesiodic Theogony. Once we get to the Roman era there are plenty of examples of Greek writers discussing Roman literature and using the Greek names. When Plutarch mentions Roman gods in the context of his biographies, he uses the Greek names. Like I said: this is translation, not a theological thing.
You last paragraph is a very different kind of question and really warrants a separate post. I hope you won't mind if I don't try to tackle it.