I have been doing this with webrole. But as I understand windows azure websites: multiple sites hosted in a single webrole.
Q.1. Is it possible to connect to webrole?
Q.2. Is it possible to atleast connect to Azure IIS management console?
No, with Windows Azure Web Sites it is a managed solution. You can edit web.config and use the management portal, but there is no more direct ways to access the underlying compute resource.