I'm not aware of a custom admin workbook where I've seen datasource connection details like what you're after.
But, there is a ERD out there of the PostgreSQL database that sits under Tableau Server and if you access some combinations of the hist_data_connections, hist_datasources and historical_events tables, you could get what you're after. (requires unlocking the readonly user on the PostgreSQL database in your Tableau Server Administration).
I'm trying to envision how I can manage access to extracts/live (particular extracts though) data sources without having to publish these extracts in multiple projects.
Is your best practice to lock permissions to the project and just publish extracts to the project even if they are the same extract?
I'm considering the "You and I work in the same department (so we share a project) and you can have access to an extract that I can not" does that equal a new project where that extract can be stored?
If it were me, I'd want to establish a project just for Published Data Sources and then use User Groups either on Tableau Server or in Active Directory and grant access and role to each source by Group.
Unlike an old OLAP cube model where "one source fits all," Tableau can result in a ton of highly customized sources for given dashboards, so this could end up being pretty involved.
I've been trying to build the connection, but there isn't a straight forward path in Postgres. It was easy linking the Data Sources to Projects, and Users to Groups. But I have yet to discover the way to link it all together.
Basically what you are looking for is a report on permissions and that has been discussed, however, it's not as easy as one would think & hope. Here's one conversation inside Matt Coles's excellent custom admin view workbooks resource, Shareable Data Sources for Tableau Server.
Here's where you can get a start: Getting permission settings from the Tableau Server
This likely won't have all of the answers you're looking for but it's the best we've got that's quick/easy to use. Coupling this with Matt's workbooks -- or the JOIN logic of his & mine -- you may be able to get some higher-level answers.