Task Manager will tell you if you haven't' turned on the show database process windows options under Options > Developer. If that is ticked then look for an icon at the bottom of the screen.
That will stay running whilst Trainz is running and will close itself once Trainz has closed - if you shut down before this happens it will affect the database.
No to the first part as that process will close shortly after you close Trainz itself depending on what it's doing at the time. However you do need to wait for that to close before shutting down.
No worries - hopefully my advice will help you reduce the chance of getting the updating asset database message. Be aware though that if TADDaemon crashes for some reason you may still get a database repair on the next startup.