If a student loses connection to our testing servers (lost internet, device power, etc.) it can be confusing for them when they attempt to sign back into the test player but cannot locate their name when they do. Students must be in the Awaiting Student status on your proctor console in order to find their name in the drop-down selections. The actions that you would choose in your open session depends on what testing status shows for that student. Read more about this with Student cannot find name when joining a test session.
If a test was suspended and you wish to have the student complete their test at a later time, you can either use the same test session that they started in or just create a new one for the suspended test. Tests follow students and not sessions so deleting a test session is no issue at all. Read more about how to find suspended tests and those that have not started yet with How to find suspended tests or students who have not yet tested.