U r correct in that the Republic of Kalmykia is Europe's sole nation where Buddhism is an 'official' religion - however it is officially recognised as a religion throughout Russia in the Federal Constitution, together with Russian Orthodoxy, Islam and Judaism. About 50% of the population there r Mongol Kalmyks and Buddhist with some Shamanism. Unfortunately their language is dying out, as a result of Stalin's national deportation and resulting genocide in 1944 (together with most Caucasus minorities).
U r correct that Buryats (likewise Mongols) are also Buddhist (and often also Shamanist), however the Russians r a big majority in that republic. Buryatia has a very small minority population of close relations of the Turkic Tyvans, on the common border.
As for the Republic of Tyva, it is 1 nation with 2 spellings of its name, Tyva, formerly Tuva, which may have led u 2 think it's 2 places! About 75% r Tyvans and Buddhist/Shamanist, the sole Turkic people 2 b so. Their Turkic language is the reason for their split from Mongolia after the Chinese Revolution in 1912. Since they were only annexed by the USSR in 1944 their language remains in daily use. Some Tyvan villages lie in Mongolia and in China.
Buddhism has started 2 grow in the (Turkic) Republic of Altai, 2 the West of Tyva, with monks trained in other Buddhist centres in Russia and further abroad.
St Petersburg has a large temple dating from tsarist times, while Elista, the capital of Kalmykia, has Europe's largest temple and Buddha figure.
As usual, Wikipedia has much more detail of interest.