what I\'m trying to accomplish is to run commands inside of a Docker container that has already been created on a Digital Ocean Ubuntu/Docker Droplet using Ansible.
You should be able to execute a script (with your sequence of command in it) with docker exec:
docker exec container-name bash -l -c /path/to/script > /path/to/log
(See also "Why do I have to use bash -l -c inside my container?")
/path/to/script
should be accessible by your Ansible process. /path/to/log
is a path inside the container, that could be shared in a volume.