1. DEEP is designed to be Platform-as-a-Service for serverless environments that developers and businesses could use in their own accounts. AWS Lambda is one of those services. The fact that right now only AWS provides the full spectrum of services made us build it on AWS first.
2. "Sandstorm makes it easy to run your own server" and "OpenShift gives you all the tools you need to develop, host and scale your apps in the public or private cloud" - so I would say neither, because the goal as you further described is to be able to support cross-cloud providers, while above ones are service-specific (am I wrong?)
3. You are correct. Right now it's an abstracted layer on AWS, and we are planning to add in near future more providers like Azure, Google Cloud and more. For now, it makes the developer job easy by abstracting everything except application layer, and have it scaled at AWS size, which is virtually infinite.
Does it make sense? And to go even further, what would you improve in our documentation? :)