Just posting a link to a discussion which might be an answer for timing your students.
https://moodle.org/mod/forum/discuss.php?d=262935
Hope that helps?
As for the Logout screen - unless you have added a logout screen there isn't one in the More themes, that I recall, but there is one in Moodle that pops up now and again where it asks you, "Are you sure you want to logout?" If that is the page then it can be fixed, I just need to know how to trigger it so I can get the correct CSS for it!
Cheers
Mary