The tracker is not visible for you, if you are included in a user group, which does not have the permission to see the Tracker.
You can log time for as many users as there are available in your system.
It is only possible to log times for users, that have the needed permissions to log time on the issues.
Yes, the tracker keeps on going, until you press pause or log your work session.
You will be able to extend your testing phase up to 3 times per account per app.