BuddyPress.org

Did you enable pretty permalinks like mercime suggested? Are you using the latest version of BP?

Did you install WordPress in a subdirectory with these instructions?
https://codex.wordpress.org/Giving_WordPress_Its_Own_Directory

If so, there are bugs with BP 1.2.8 relating to subdirectory installs.

Please read this thread to resolve the login/logout issue for now:
https://buddypress.org/community/groups/creating-extending/forum/topic/loginlogout-from-admin-bar-does-not-work