Disclaimer : This answer is correct as per Nov 2016.
Yes you can use it for free. But some people use the paid one because they want a private docker hub, or they prefer it in their own premise.
No. but using Vagrant makes it easier for you to follow the tutorial
[UPDATED] Yes. Dockerhub introduced automatic build. Never use it though. In my workflow, I use jenkins to build my dockerfile. You can follow this sample workflow (using jenkins ) from docker.
Just start it when you are starting your development. Anyway one of the benefit of docker is that you able to eliminate issue where it is working in dev environment but not in the production environment.
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