How aware are Russian people about population transfer in the Soviet Union in general, and what opinion do they have about such actions?
In the last years of Soviet Union (and in modern Russia too) there was quite much information available on these topics. I say, much more than disinterested people in the West may and/or want to know.
Crimean Tatars were among strongest supporters of Nazi regime in occupied territories of SU. So that deportation was considered as a kind of punishment at that time; quite in line with other similar actions which took place after WW2.
Of course, such an indiscriminate measure isn't admissible by any modern standards. In 1989 the Supreme Council of Soviet Union recognized it as unlawful and criminal.
But if you ask about people's opinion in modern Russia, I'd say that there are still different points of view. That is some people still argue that "they fully deserved this". I don't think that the latter opinion is really predominant, yet it seems to have more than just marginal support.