Data is only accessible during only one session and make something after timing out

I want to make a web application that has tests/exams that are accessible during only one session.

The logic I have is that a user can enter an exam only one time if he entered it and finished it he shouldn't be able to enter it again. for that the problem is easy, maybe I can make a flag determines if he entered the exam or not.

But what makes it complex that I want him to be able to access the exam if his session is still alive even if he got out of the exam to do something and come back as long as his session is not expired. that's one thing I don't know how to handle?

And if he closed the browser and his session is timed out I want to send him an email contains the exam's result but how will I check the session while there are no further requests from the user (as he already closed my website)?

I use node.js but the idea is what concerns.