Yes, in the self-hosted version you can "bring your own database". Right now the choices of databases is limited though as only MySQL is supported. It is however planned to support more database systems in the future.
#Can I perform reads and writes directly against the database when using Graphcool?
No, Graphcool abstracts away the database layer and does not expose any direct interfaces towards the database.
No, Graphcool's GraphQL engine effectively takes the role of what would be considered an ORM in Graphcool's architecture (though technically that's not 100% accurate). It retrieves the data from the database and exposes it through the API.