What does "Nos" do in this sentence? I feel like if I take it out it would still be comprehensible and grammatically correct
Yes, subject pronouns are optional, but right now we're learning what they are and where they go when we do use them. Also, depending on how the prompt is formatted, it's the key part that tests your knowledge of verb conjugation.
The verb studeo ("strive for, study, occupy oneself with") is special in that it takes a dative object, hence litterīs Latīnīs.