There were no Russian domination in sport, if we talk about sport in general. If you google "Soviet (Olympic) champions", you'll see people of different ethnicities. Sometimes certain sports were dominated by people from certain republics (that probably depended on what sports were popular there due to cultural differences or climate). For example, wrestling champions were almost exclusively machos from Georgia, Azerbaijan, Armenia, Ossetia, etc. Ukrainians succeed in soccer, Russians in hockey, and so on.
Your teachers were smart.I remember from this time that people said, "blah, blah, blah Russians...." and teachers said "don't say Russians, say Soviets". It never really caught on though!
The USSR was a union of 15 ethnic republics, people of various backgrounds all mixed together. For example, in my school class we had loads of kids of mixed ethnicities, who were proud of their "ethnic" last names (Georgian/Armenian/Uzbeck/Arab/Korean/Karaim/Latvian and some others).
Absolutely everyone had their ethnicities marked in passports, along with name, address and some other info. Jews were no different.One thing that seems strange though, is that USSR apparently marked in people's passport who was Jewish (why??!)
If a child was of mixed ethnicity, parents could choose which ethnicity they prefer to "transfer" to their kid.