The default database is called neo4j on both community & enterprise. You also get system which contains users, roles, permissions, and so forth. It's always blank with docker, unless you mount an existing directory mapped to /data in the container, in which case you can place your own database in there.
No, 4.0 doesn't support renaming of databases. But if you're using docker you also don't need it. You can basically copy that folder to another folder of a different name, and start the DB with a different name in the default database setting, and it accomplishes the same.