One great advantage of using Azure Websites is that I can get secure HTTP (HTTPS) without doing nothing: I simply type https://xyz.azurewebsites.
https://xyz.azurewebsites.
No. HTTPS is not offered for .cloudapp.net domain as of today. Also since you don't own .cloudapp.net domain, I don't think you can buy a SSL certificate for that. If you want you could create a self-signed certificate and use that.
.cloudapp.net