Home page instead of CB Login after session timeout ?

Hi !
I do not know if this behavior comes from CB or Joomla or my settings.... 
After idle and session time out in registered state, when user wants to reactivate, it goes to CB login page. It is logic.
However, I will be interested that users goes to homepage instead. 

furthermore, it seems that user is still in the list of "connected" even if the session was terminated 

Where they go to depends on where they were when their session expires. If they were on a page that requires user access it's going to send them to login page not the home page. There's no changing that.

furthermore, it seems that user is still in the list of "connected" even if the session was terminated

They still have a session it's just not logged in anymore. It takes awhile for that to clear out as it'll clear when session garbage collection happens.