Looker does not support running in Docker containers, but I do have a Docker setup on my local machine for testing our latest Looker releases. All I need is a fresh install with the looker-latest.jar file from our S3 bucket, with no saved config information. Docker is cool for this.
As you mentioned there are a number of issues running Looker in a container that negate many (most?) of the benefits of containerizing things, namely that there are lots of Looker things that need to be persisted between container destruction/construction. They include the things you listed along with model* directories, ssh/cluster-shared directories, tunnel configurations, caches and things that are added to the filesystem while you're doing work (think Big Query certificates, for example).
One big benefit of Docker is that you can create ephemeral, lightweight services that you can automagically scale up/down. The Looker java application doesn't really fit within this paradigm.
If you're using Docker for Looker release testing or your own unit testing that may be appropriate. Using Looker in Docker in production is a risky proposition at best, and 'is not supported.'