I have played several of the scenarios in C3C but at the moment a scenario is finished (victory / defeat /number of turns) some of the end scenes are shown and then the game crashes. None of the finished scenarios are saved or shown in the list of finished games.

What is happening?

Could this have something to do with he fact that i play this logged in as a user with restricted acces and not as the administrator?

That's a possibility. Due to some internal design decisions, you need to be an Administrator to play Civ3.