Implement on-demand docker container start-up
You could use Kubernetes' built-in Horizonal Pod Autoscaling (HPA) to scale up from 1 instance of each container to as many are needed to handle the load, but there's no built-in functionality for 0-to-1 scaling on receiving a request, and I'm not aware of any widely used solution.
You can use systemd to manage your docker containers. See https://developer.atlassian.com/blog/2015/03/docker-systemd-socket-activation/
Some time ago I talked to an ops guy for pantheon.io about how they do this sort of thing with docker. I guess it would have been before Kubernetes even came out. Pantheon do drupal hosting. The way they have things set up, every server they run for clients is containerised, but as you describe, the container goes away when it's not needed. The only resource that's reserved then other than disk storage is a socket number on the host.
They have a fairly simple daemon which listens on the sockets of all inactive servers. When it receives a request, the daemon stops listening for more incoming connections on that socket, starts the required container, and proxies that one request to the new container. Subsequent connections go direct to the container until it's idle for a period, and the listener daemon takes over the port again. That's about as much detail as I know about what they did, but you get the idea.
I imagine that something like the daemon that pantheon implemented could be used to send commands to Kubernetes rather than straight to the Docker daemon. Maybe a systemd based approach to dynamically starting contaners could also communicate with Kubernetes as required. Either of these might allow you to fire up pods, not just containers.