Does Kubernetes mount an emtpyDir volume on the host?

Yes it is also accessible on the node. It is bind mounted into the container (sort of). The source directories are under /var/lib/kubelet/pods/PODUID/volumes/kubernetes.io~empty-dir/VOLUMENAME

You can find the location on the host like this:

sudo ls -l /var/lib/kubelet/pods/`kubectl get pod -n mynamespace mypod -o 'jsonpath={.metadata.uid}'`/volumes/kubernetes.io~empty-dir