Can I add, rename, or delete ticket statuses?

Yes, ticket statuses live in an editable workspace list named "Ticket statuses" (described as "Statuses for tickets"), which is pre-populated with Open, Pending, Resolved, and Closed. Deleting a status requires an Admin or Owner role; users with the User role get a "No access" error. The default status for new tickets and the statuses configured as hidden by default (Resolved and Closed out of the box) cannot be deleted.