Key rules
Russia
You need documents proving legal stay and registration, the child's fingerprinting, a medical certificate, SNILS and TIN. The test is oral and written (first-graders take only the oral part), and you can retake it no earlier than after 3 months. Whether children from EAEU countries must be tested and which certificates are required, check with the school or the local education authority. The rules change, so check the MVD website and Gosuslugi (Госуслуги).
Sources: Ministry of Education of Russia («Минпросвещения России»), Gosuslugi («Госуслуги»)