And finally concerning powerbases, Traya has the assets of Nihilus and Sion because they were under her direct command and therefore their assets were her assets. The Ravager, the Sith fleet, the assassins etc. they all belonged to her as the Executor belongs to Sidious. However this does not include things that came into their ownership post-betrayal e.g. the Tomb of Freedon Nadd, war beasts, Vaklu, etc.
This is not really true. The group was called the Sith Triumvirate. A triumvirate is a government co-ruled by three heads of state. Each three of these had there own powerbases, agendas, and schemes. In fact, a Kaggath could be held between the members of this triumvirate. Traya was not "in direct control". The others cooperated with her and her plans, but as seen with their betrayal, they obviously were able to resist her command. Notice that its a betrayal, not a mutiny. Therefor Traya should only have her powerbase, or the forces of Malachor V.