You can get at a list of published reports and some schedule-related information through Tableau Server's admin views:
There is no way to (easily) explore the SQL that might be fired by individual reports, however. You'd need to experiment with the reports locally to see what they do.
Is it possible to query the PostgreSQL database itself?
We are needing to find things that are in the metadata.
For example: We were notified today that all reports that refer to our "VU" campus, should be updated to refer to that campus as "Online". In our ERP the campus code can be any number of values, so we have a cross-walk table to help us group the various campus codes from our ERP as "VU". We could update our cross-walk table to use "Online". Unfortunately the fix is not that easy.
In many reports there is can be further groupings, based on the audience for the report. The additional grouping might have been done with CASE statements in the SQL script itself, or it might have been done as a calculated field. To find all the SQL scripts that might have a CASE statement that refers to the "VU" campus and all the reports that might have used a calculated field for grouping is a major undertaking, if we have to look at every script and every worksheet.
This is just one example of several things we are needing to find/change/track.
If we can query against the PostgreSQL database to find specific strings in specific areas it would be extremely helpful and time saving. As well as give us a method to document/track and fix standardization issues and changes to those standards.