Just a guess, but a possible reason (for, at least, the logging-in bug) is that the log-out time after being inactive (which was normally about 10 minutes for this forum) has been changed to 0 seconds, perhaps by accident or a bug. Checking "remember me" prevents timed log-out. Also, logging out clears your forum cache ("all cookies cleared!"), resetting unread posts among other things. Every time you load a page, you've already been logged out, but the forum "remembers" you so you invisibly log-in again - but your cache has already been cleared in the process. And that's why clicking New Posts never works. And other users can't be seen on Currently Active Users, because they're actually logged in only for the fraction of a second whenever they interact with the forum.
Of course, this might all be wrong. :-P I don't really know how VBulletin works.