Typically, an exchange student is enrolled in the host faculty in which he plans to study. Attending 1-2 additional courses at another faculty is possible but it needs to be mutually agreed in advance.
Most courses are conducted in Russian, some in English. Please check the language of instruction in course catalogues (please see below).
Please note that if you want to take a subject course that is conducted in Russian, you must have a TORFL-II / B2 level of Russian language. If you want to take a course that is conducted in English, you must have the level of English which is equivalent to IELTS 6.0.