I think the best approach for your cert issue is to abstract that into a separate service (nginx is an option, but I'd recommend the Rancher approach below). Yes, that means you have to add another container, but that's just another block in your docker-compose.yml file. Embrace the approach of separating your components into containers and organizing them as a stack. You can easily link containers together, share data volumes, and start/stop individual containers or the stack as a whole.
The problems that you're having are pretty easy to fix with some tooling. Rancher (http://rancher.com/) greatly simplifies the cert issue by allowing you to import certs and provide them to the Rancher loadbalancer service (which you can add to any stack). There's also a LetsEncrypt community catalog template that automatically retrieves and imports certificates to Rancher. There are other open source orchestrators like DCOS, but Rancher is probably the simplest to use, and it's the only one I'm very familiar with. There are SaaS options that you can look into, but I don't have experience with them.
As for the container crashes, it's trivial to automatically restart them. Just pass the --restart=always flag to the Docker run command. You can also add the flag to a docker-compose.yml file.