There is an extension which will record exports using civicrm export from contact search. Unless you have this extension installed there really isn't a way to see if there were specific exports of information.
More information on the extension can be found here
Specifically for "what records have they exported", I don't think there's an easy answer to that one. Most sites, not just civi, don't track outbound data.
The more general question is more of a web hosting question. For drupal specifically if you have the built-in statistics module enabled you can go to /admin/reports/hits and mostly see what pages they ...