Normally, to show my clients work in progress, I put their rails app in a sub domain of my website. Is that possible with digitalocean or will the subdomain need to be started in a new droplet ?
It certainly is possible. You can add a new virtualhost that handles the subdomain: How To Set Up Apache Virtual Hosts on Ubuntu 14.04 LTS. If you're using unicorn, you will need to configure an additional instance that serves the new app.
Yes, that's fine. In that case, you can just add an nginx virtualhost with the appropriate passenger_* directives:
How To Set Up Nginx Server Blocks (Virtual Hosts) on Ubuntu 14.04 LTS.
