Your summary is almost right but it does not reflect the essence of what mesos represents. The vision of mesosphere, the Company behind the project, is to create a “Datacenter Operating System” and the mesos is the kernel of it in analogy to the kernel of a normal OS.
The API is not limited to Java, you can use C, C++, Java/Scala, or Python.
If you have set-up your mesos cluster, as you describe in your question and want to use your resources, you usually do this through a framework instead of running your workload directly on it. This doesn’t mean that this is complicated here is a very small example in Scala which demonstrates this. Frameworks exist for multiple popular distributed data processing systems like Apache Spark, Apache Cassandra. There are other frameworks such as Chronos a cron on data center level or Marathon which allows you to run Docker based applications.
Update:
Yes, mesos will take care about the placement in the cluster as that’s what a kernel does — scheduling and management of limited resources. The setup you have sketched raises several obvious questions, however.
Layers below mesos:
Installing mesos on CoreOS is possible but cumbersome as I think. This is not a typical scenario for running mesos — usually it is moved to the lowest possible layer (above Ubuntu in your case). So I hope you have good reasons for running CoreOS and a hypervisor.
Layers above mesos:
Kubernetes ist available as framework and mesosphere seems to put much effort in it. It is, however, without question that there are partly overlapping in terms of functionality — especially with regard to scheduling. If you want to schedule basic workloads based on Containers, you might be better off with Marathon or in the future maybe Aurora. So also here I hope you have good reasons for this very arrangement.
Sidenote: Kubernetes is similar to Marathon with a broader approach and pretty opinionated.