1.USSR didn't have state property. It had public property. The difference is you had to explain to the people why you are taking their property away.

2.Each republic made a different decision. Latvia, for instance, went for the restitution of pre-Soviet owners. Belarus left the things as they are. Russia did privatization which caused enormous troubles since the parliament did not like that but the president did. Since the POTUSRF was also commander-in-chief he just had the parliament shelled and turned Russia into a Presidential Republic, like the USA and France. Previously it was a Parliamentary Republic, like Germany.

Why are so many Russians rich? That part isn't any different from Saudis or Texans: They are rich, because oil.