How to handle DB updates when my Drupal source is in git?
Yes, if you update the files on the production site, all you have to do is to run the update.php or drush updatedb additionally.
For migrating functionality from staging to production site, you might be interested in Features. It allows you to export various "pieces" of functionality and transfer them to another installation; this can include content types, fields, views and image styles, to name a few.


Hope this helps!