If you use the "remember me" feature, you will be logged out automatically after the same timeout, but the session will resume next time a request is made, effectively keeping you logged in. So you shouldn't experience a logout when checking "remember me".
As for the auto logout timeout, it was recently discussed here https://discuss.flarum.org/d/21562-login-session-timeout
Unless Flarum periodically pings the backend, I don't really see an option to show you have been logged out. (Plus, pinging would keep your session alive, which might or might not be desirable). Pusher could be used, but that would also require a job in the backend to trigger when your session expires, which is not possible without setting up something like a cron job.