How to execute a Bash command only if a Docker container with a given name does not exist?

You can check for non-existence of a running container by grepping for a <name> and fire it up later on like this:

[ ! "$(docker ps -a | grep <name>)" ] && docker run -d --name <name> <image>

Better:

Make use of https://docs.docker.com/engine/reference/commandline/ps/ and check if an exited container blocks, so you can remove it first prior to run the container:

if [ ! "$(docker ps -q -f name=<name>)" ]; then
    if [ "$(docker ps -aq -f status=exited -f name=<name>)" ]; then
        # cleanup
        docker rm <name>
    fi
    # run your container
    docker run -d --name <name> my-docker-image
fi

I suppose

docker container inspect <container-name> || docker run...

since docker container inspect call will set $? to 1 if container does not exist (cannot inspect) but to 0 if it does exist (this respects stopped containers). So the run command will just be called in case container does not exist as expected.

You can use filter and format options for docker ps command to avoid piping with unix utilities like grep, awk etc.

name='nginx'

[[ $(docker ps --filter "name=^/$name$" --format '{{.Names}}') == $name ]] ||
docker run -d --name mynginx <nginx-image>