First. Yes, there was inheritance in the USSR. In the early years of the Soviet power there was a limit on the cost of property that could be inherited. This limit was canceled in 1926. The rules of inheritance changed few times during the Soviet period, mainly towards widening of the rights of testators and their heirs, simplification of the procedure of inheritance. The last edition for the inheritance law was made in the 60s and this law was in act till 2002. You can read about the history of inheritance in the Soviet Union here 6.
Second. No, there was no such thing as giving back the properties of the pre-Soviet owners. There was a lot of talk about it but the state didn't do anything as far as I know.