Docker containers are generally made to host a single service, thus, they don’t really use systemctl
Some containers do support systemd, in case you’re running multiple services on a single container, is that the case though?
For the status, you can define a healthcheck script that runs every certain interval and tests your service.
As for if the process is running, the container only runs so long as the process is running, so assuming your container is running, the process is as well