We received this list at the International Exchange Center when applying for the program, but Marina (our coordinator) warned us right away that the South African embassy is a bit strange and often asks for something new. We confirmed this later on, but here are the details about the documents:

1. Application form - nothing complicated here, just another form in English :))))
2. 1 passport size photo - a passport-style photo; you will need 2 more later for the embassy.
3. Copy of passport - international passport + national passport (a copy of the registration page is mandatory).
4. Letter of motivation - time to use your imagination and write about how much you want to see penguins and get to know such a beautiful country as South Africa))) and don't forget to have someone check it, since it's all written in English.
5. CV - resume, write about how wonderful and hardworking you are, also in English.
6. Proof of student status (full time students) (if graduated copy of Diploma not later than half a year after graduation) - actually, I study in the evening department, so I asked them to just state 'full time student in evening' in the translation, and this certificate must also include the university graduation date.
7. Medical report - this is a very funny document. The form is provided by the International Exchange Center; it has a huge list of diseases and is all in English. You need to find a place where they can stamp it to confirm you aren't sick... yeah... doctors wave their hands when they see it =)
8. Copy of the police clearance report - this is one of the longest documents to get, it takes from 3 weeks, so as soon as you decide to join the program, run to get this certificate! You don't need an apostille, a translation is enough.
9. Insurance details - insurance, done at the International Exchange Center, costs about $300.
10. Bank statement about sufficient funds - a bank statement for the last 3 months, valid for 2 weeks, and it's only needed at the embassy. I want to note that the statement can only be from your own account or your parents' account... Meaning neither a brother, sister, nor grandmother can be your sponsors...
    Also, you need to provide a 2-NDFL form (also valid for 2 weeks) to the embassy, flight reservations, and at the time of visa issuance, you need to bring purchased tickets and a sponsorship letter (in English).

I hope I didn't forget anything... It took me almost 6 months to collect all these documents, but I wasn't in a hurry.

Thanks to Marina for not letting me give up in early September and cancel the trip!