The current document is directed to an interface and authorization service that allows users of a cloud-director management subsystem of distributed, multi-tenant, virtual data centers to extend the services and functionalities provided by the cloud-director management subsystem. A cloud application programming interface (“API”) entrypoint represents a request/response RESTful interface to services and functionalities provided by the cloud-director management subsystem as well as to service extensions provided by users. The API entrypoint includes a service-extension interface and an authorization-service management interface. The cloud-director management subsystem provides the authorization service to service extensions that allow the service extensions to obtain, from the authorization service, an indication of whether or not a request directed to the service extension through the API entrypoint is authorized. Requests for service-extension URIs within the API entrypoint are processed by a cloud API entrypoint server that dispatches requests, in a predetermined order, to service-extension servers.