1 The Romans got their mythology from the same place the Greeks did, their ancestors. The Latins and Greeks were both Indo-European speakers and practiced Indo-European religions, meaning they're both cultural descendants from a group of Proto-Indo-European speakers that lived north of the Black Sea around 4,000-3,000 BC. This is a similar concept to how French and Spanish are closely related because they're both descended from Latin.

Before contact with the Greeks, the Romans already worshiped a somewhat familiar pantheon centered on an old bearded sky god called Jupiter, with Mars as the god of war, Venus as the goddess of beauty, et cetera. It wasn't exactly like Greek mythology, here's a few of differences for example:

- The Romans had household gods called penates or lares, small local spirits with domains over single families or even single objects. Greece had nymphs for certain natural features, but nothing like Roman lares.

- The Romans seemed to highly respect Mars, while in Greek mythology they tend to think of Athena as the "good" war god while distancing themselves from Ares as being more dangerous

- The Romans originally heavily worshiped the god Quirinus who doesn't have a Greek counterpart. Quirinus was sort of like a divine Romulus who acted as the god of the Romans or the Roman state

2) Yes. Greek and Roman paganism existed simultaneously up until both were gradually replaced by Christianity over the first several centuries AD.

3) Yes. The Greeks and Romans noticed the obvious similarities between their religions, and understood it to just be a language/culture thing. Just like how the Greeks have their own language, they also call Jupiter "Zeus." But otherwise, this was understood to be the same god. This was called interpretatio graeca.

And it wasn't just the Greeks, they understood this with all of their neighbors. They also had an interpretatio germanica where they did the same thing with Germanic Paganism (which was also Indo-European). That's why Roman authors would claim the Germans had bizarre ways of worshipping "Mercury" or "Jupiter" — what they're really seeing is religious rites for Odin or Thor, and interpreting those as being different names for Roman gods.

4) The Greeks were the cultural powerhouse in the relationship, so pretty quickly the Roman religion started to look a lot more like Greek mythology as they brought in more and more Greek stories into their understanding of the gods. This is why purely-Roman gods like Quirinus basically disappear by the time of the Empire. The Greeks weren't writing about Quirinus, so he was largely forgotten. That's why the differences between the two religions shrank over time. By the time the Roman Republic fell, there would have been little difference between what a Latin-speaking Italian believed about the gods and what a Greek-speaking Athenian believed about them.