In my Latin class, we are constantly learning about the history and culture of Rome. Do studies show that this helps with language learning or is it unnecessary?
Yes. Actually, it is quite impossible to not to.
In this article, it states:
The link between foreign language learning and culture learning has been established by the linguists and anthropologists a long time ago. The American Council on the Teaching of Foreign Languages has concluded that through the study of other languages, students gain a knowledge and understanding of the cultures that use that language. Moreover, students cannot truly master the language until they have also mastered the cultural contexts in which the language occurs. Linguistic competence alone is not enough for learners of a language to be competent in that language. From simple, everyday things, like forms of address to appropriate ways of expressing disagreement, culture forms an integral part of the language learning curricula. In any case, in order for communication to be successful, language use must be associated with other culturally appropriate behavior, not only linguistic rules in the narrow sense.
Yes that is a lot. But in short, the relationships between the culture and the actual language have been known for a very long time. By learning the language, students also learn about the cultures that use the language. So, by learning the culture, you can get a better understanding of the language and vice versa. It is practically a win-win. Culture can be used in your language anytime, from simple tasks to speaking with the leader of your country. Language should not only be used by itself, it should be slightly influenced by the appropriate culture.
Since people of a like language will naturally be more in communication with each other than, say, those who don't share a language, and thus forming a culture, the habits, concepts, and even words and their real/implied meanings are going to be deeply intertwined with that culture.
This document points out:
Language is more than just the code: it also involves social practices of interpreting and making meanings
There is a fundamental relationship between language and culture
Understanding the nature of the relationship between language and culture is central to the process of learning another language. In actual language use, it is not the case that it is only the forms of language that convey meaning. It is language in its cultural context that creates meaning: creating and interpreting meaning is done within a cultural framework.
with regards to learning a dead (nonspoken) language such as Latin, learning the culture may not help as much as it might with other languages. See the first paragraph in this answer:
most people learning dead languages aren't learning to speak, they're learning to study anthropology, literature, history, or some other field that doesn't require speaking the language.
I'm not saying that studying the culture is useless in the context of learning a dead language; it will still be helpful for interpreting texts. When actually speaking a language, though, it is vital that you know what you're actually communicating, and the language's cultural quirks will drastically affect that.