ERROR: Cannot create container Conflict. The container name is already in use by container "". You have to remove (or rename) that container to be able to reuse that name. code example

Example: docker compose Conflict. The container name

Your case is probably related to a bug that will be fixed in the 18.03 release. Some workarounds are proposed here:

https://github.com/docker/for-linux/issues/211
https://github.com/moby/moby/issues/36145
docker-compose up builds, (re)creates, starts, and attaches to containers for a service.

Since your images are built and the containers of your service have started, you can then use

docker-compose stop and
docker-compose start
to start/stop your service. This is different from docker-compose down which:

Stops containers and removes containers, networks, volumes, and images created by up.

Regarding the danger of losing your data if a container is removed, read about persistent storage and how to use volumes.