There is an option to enable mounts while starting the Hyperkit VM with the argument --mount. This can be used with minikube start command. It would look something like this (based on the manual, haven't tried this myself).

minikube start --mount=true --mount-string="/myvolume:/test"

With this it should be always available.

About the second question, you are right docker daemon comes from minikube so a K8S cluster is required. If you only need Docker API and daemon (not K8S) then I suggest to continue with Docker Desktop. If for any reason you don't wish to use Docker Desktop (license etc), then I suggest looking into LIMA and nerdctl. I haven't tested them out yet but they look promising. Check out this blog article to get started: https://medium.com/nttlabs/containerd-and-lima-39e0b64d2a59