How can I modify what is displayed on the civicase dashboard?
One way would be to use the technique described here:
http://wiki.civicrm.org/confluence/display/CRMDOC/Customize+Built-in%2C+Profile%2C+Contribution+and+Event+Registration+Screens

The files you want to override are:

    templates/CRM/Case/Page/DashboardSelector.tpl
    and
    CRM/Case/Page/Dashboard.php


Getting custom fields can be done using the api, although you need to know the database id for the field definition. See http://wiki.civicrm.org/confluence/display/CRMDOC/Using+the+API, and down where it says Custom fields.