As a sidenote, how did the USSR change Russian culture? I've understood that the Soviet state tried to create a specific Soviet culture and root out national cultures. Did that apply also to the Russian culture or was this Soviet culture modeled after basis of Russian culture?
You are opening a huge can of worms here. There is no single Soviet experience. Generally speaking the 1920s were a golden era of culture within the USSR. Stalinism saw the crushing of most national cultures and near eradication of some peoples due to deportations: Chechnians, Crimean Tatars and Meskhetian Turks for example. Krushchev's thaw allowed non-Russian cultures to come back out into the open a bit. By the time of Brezhnev russification was the norm, but more passively than under Stalin - people weren't being shot for being nationalists, but opportunities really only existed for Russian speakers. The late 80s saw the flare up of nationalism: Nagorno-Karabagh, Osh conflict in '89 and demonstrations in Almaty.
Diaspora types have their axe to grind against the Soviet Union, but it is hard to dismiss the cultural achievements of Soviet citizens like Shostokovich, Bulgakov and others. The ossified émigré culture of pre-revolutionary Russia represented something that was horrendously francophied and maybe 5% of the population, at most, participated in. The USSR made it so that most of the population had the ability to be part of Russian culture, so out of necessity there were changes. But I'm not sure I'd call all of them negative.
For some cultures the USSR was also a generally positive thing (for example Kyrgyz culture) as the USSR created a standard written language for the first time, prompted creative development in said language, opened schools and universities in such languages and left a solid foundation for these cultures to develop independently now. Others places are a mixed bag, such as Ukraine. The 1920s were a huge boon to Ukrainian culture, but the repressions of the 30s and later russification certainly damaged Ukrainian culture in Eastern Ukraine. On the otherhand, were it not for the Soviet led ethnic cleansing of Western Ukraine and forced ukraininization, the major cities of Western Ukraine wouldn't be Ukrainian speaking today. Other cultures were nearly wiped off of the map by Stalinist repressions or by a more passive russification. Belarus' is an example of the later.
So to answer your question - yes Russian culture changed because of the USSR, but these changes were inevitable as Russian culture moved from a gentry culture to modern national culture. Socialist realism was constraining, but brilliant works were still produced under it.