Does Azure offer https for "cloudapp.net"?

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.

I would walk through the documentation listed here: http://azure.microsoft.com/en-us/documentation/articles/cloud-services-configure-ssl-certificate/

Since you're getting a timeout with HTTPS (rather than a certificate error), check that you have a HTTPS endpoint defined in ServiceDefinition.csdef.

Additionally, be aware that the redirect-to-subdomain approach isn't much more secure than using a self-signed certificate. The reason browsers reject self-signed certs is that they are vulnerable to spoofing attacks: a user can't detect if an attacker has, for example, hijacked the DNS to point to his IP address instead of yours, where he hosts a facade of your site that just collects passwords or whatever.

In your scenario, the cloned site could redirect to another a second clone, one that is a facade of your cloudapp.net site. It could be even be secured with the attacker's SSL certificate. Unless the user was trained to recognize the host name of the real cloudapp.net, she wouldn't know she was on the attacker's "secure" site.