How does Docker handle different kernel versions?
Docker's behaviour is no different: it doesn't concern itself (directly) with the behaviour of the containerized process. What Docker does do is set up various parameters (root filesystem, other mounts, network interfaces and configuration, separate namespaces or restrictions on what PIDs can be seen, etc.) for the process that let you consider it a "container," and then it just runs the initial process in that environment.
The specific software inside the container may or may not work with your host operating system's kernel. Using a kernel older than the software was built for is not infrequently problematic; more often it's safe to run older software on a newer kernel.
More often, but not always. On a host with kernel 4.19 (e.g. Ubuntu 18.04) try docker run centos:6 bash. You'll find it segfaults (exit code 139) because that old build of bash does something that greatly displeases the newer kernel. (On a 4.9 or lower kernel, docker run centos:6 bash will work fine.) However, docker run centos:6 ls will not die in the same way because that program is not dependent on particular kernel facilities that have changed (at least, not when run with no arguments).
Let's say that I make an image for an OS that uses a kernel of version 10.
I think this is a bit of a misconception, unless you are talking about specific software that relies on newer kernel features inside your Docker image, which should be pretty rare. Generally speaking a Docker image is just a custom file/directory structure, assembled in layers via FROM and RUN instructions in one or more Dockerfiles, with a bit of meta data like what ports to open or which file to execute on container start. That's really all there is to it. The basic principle of Docker is very much like a classic chroot jail, only a bit more modern and with some candy on top.
What behavior does Docker exhibit if I run a container for that image on a host OS running a kernel of version 9? What about version 11?
If the kernel can run the Docker daemon it should be able to run any image.
Are there caveats?
As noted above, Docker images that include software which relies on bleeding edge kernel features will not work on kernels that do not have those features, which should be no surprise. Docker will not stop you from running such an image on an older kernel as it simply does not care whats inside an image, nor does it know what kernel was used to create the image.
The only other thing I can think of is compiling software manually with aggressive optimizations for a specific cpu like Intel or Amd. Such images will fail on hosts with a different cpu.
This sounds like it doesn't matter what kernel version the host is running beyond that minimum. Is this true?
As long as your kernel meets Docker's minimum requirements (which mostly involve having the necessary APIs to support the isolated execution environment that Docker sets up for each container), Docker doesn't really care what kernel you're running.
In many way, this isn't entirely a Docker question: for the most part, user-space tools aren't tied particularly tightly to specific kernel versions. This isn't unilaterally true; there are some tools that by design interact with a very specific kernel version, or that can take advantage of APIs in recent kernel versions for improved performance, but for the most part your web server or database just doesn't care.
Are there caveats?
The kernel version you're running may dictate things like which storage drivers are available to Docker, but this doesn't really have any impact on your containers.
Older kernel versions may have security vulnerabilities that are fixed in more recent versions, and newer versions may have fixes that offer improved performance.