So I know that nobles and intellectuals in other areas of Europe spoke Latin, but did medieval Russian nobles speak it given that they were Eastern Orthodox? If not, did they speak Greek, Old Church Slavonic, or just their native Slavic language? And how did this change over time?
I think this is a misunderstanding. Most government officials and religious leaders did not speak Latin in their every day lives during Middle Ages. Most spoke their local vernacular, even in places like the various kingdoms of what is now Italy. After the decline of the Western Roman Empire, Latin quickly fell out of fashion as a spoken language. However, Latin did remain the WRITTEN language of choice across much of Europe for religious and legal purposes until pretty close to the Enlightenment era. It remained as such primarily for transparency across regions where the spoken languages differed (very similarly to how English has become a common base today). It was simply easier for everyone to maintain Latin as a common base than not (by everyone, I strictly mean the government and religious institutions, as they were often the only people who knew how to read or write at all). And even the Byzantine empire, despite writing primarily in Greek for most of their existence, sometimes wrote their official records in Latin for the same reason, but primarily the records that dealt with external affairs (such as pacts, treaties, etc).
Latin never permeated into Russia primarily due to the simple fact that they were never a Latin territory to begin with, neither spoken nor written. While Greek was sometimes used in the writings of the Orthodox Church documents in what is now modern day Russia, it was Old East Slavic that dominated the written and spoken “Russian” world of the Late Middle Ages, and various other languages before that.
Edit: Grammar
Latin was studied in school in 19 and earlt 20th century. It was mostly used for study of the classics. Church Slavonic was used as written and sometimes as spoken language among the educated elite, most of which was often clergy. Modern Russian is heavily influenced by Church Slavonic. Eastern Orthodox church permits using local languages for liturgy. There was little incentive even for highly educated clergy to learn Latin. If you come to Russian Orthodox church today almost all of the service will be performed in Church Slavonic. Study of classics in original languages came through European influence after Peter the Great. As it was a standard in Western Europe of the time.