You can save the filters by embedding in an IFRAME then having some custom code setting the filters from outside the frame.
Saving the rows could be done using a hyperlink on each row to launch some more custom code to write back to your database.
That’s two great ideas - do you know if there was a dynamic user-related attribute that I could add to the URL so I know which metabase user is logged in for that session i.e. which user favorited it?
When you have Metabase embedded in an iframe I’d say it’s “the other way around”. It’s the responsibility of your (possibly very thin) outer app to authenticate the user and map that identity to a database identity through a “locked parameter” as described in the Embedding charts and dashboards with locked parameters section of https://metabase.com/docs/latest/administration-guide/13-embedding.html