The conversations tab is the same as the former private messaging and is shut off. The alerts tab is a new function which alerts the member to any new threads they are following as well as other alerts like trophies given etc.
The logout link is located by clicking on the member's name > an account dropdown appears and the logout link is the bottom bar. From what I have read, the logout button used to be in a different place in previous versions of this software but was changed to this position to be the same as facebook and google logouts. Apparently this is the preferred intuitive style.
I'll investigate further and see if the conversations tab is easily removable and if there is a workaround for adding an additional logout link.