Is it possible to read and write comments on each post for patrons only using the Patreon plugin? Like, if a reader is logged in with Patreon on the site, she can read and write. But if she’s not, a message like “Login with Patreon to comment” appear.
Currently not, but this would be a possibility with the upcoming premium addon for gating comments. Alternatively, if you want to lock your entire website to be patron-only, you can use Patron Plugin Pro - that has login lock that makes it so that only patrons will be able to register or log in.