Azure has its own object storage mechanism (similar to Swift in OpenStack or S3 in Amazon AWS) so to allow you to learn or test behavior they have an emulator so that you can mimic that storage locally.
In that case, it'd be a simple matter of tossing the old DC and spinning up a new one. More of an annoyance than anything else.
Yes. Mostly stateless. Would be scary if you put all DCs on azure though. Or any live/live system like this where an environmental change might lock out the entire environment.
I have my entire environment on AWS, spread across two geographic regions with a site-to-site VPN. Upping it a level would be putting one part on AWS and the other on Azure, with a site-to-site VPN between the two.
Yes. Having one node in Rackspace would have protected against this.