Ideas for implementing a shift scheduler? calendar? custom content type?
Maybe you are aware of the calendar module https://drupal.org/project/calendar
It has a very nice integration with views, and you could create a calender from template for a specific content type that has date fields in seconds (that generates also a view of it per week).
Of course for the case you have described you will need custom coding, surely the Drupal Ajax Framework and some other stuff... but that sounds more like fun :)