One of WordPress' biggest failings is not providing better logs. You can investigate implementing your own logging by using WP hooks and actions.
Factor #10: Dev/prod parity. Keep development, staging, and production as similar as possible. It’s all about automating your setup.
Factor #7: Port binding. Use Nginx + PHP-FPM in production for more flexibility and much better performance over Apache + mod_php.
Twelve-factor processes are stateless and share-nothing. Issues you'll run into with making sure WordPress is stateless: sessions and uploaded files.
Factor #5: Build, release, run. You shouldn't be deploying code to production if you can't reliably rollback a deploy.
Factor #4: Backing services. The important outcome of treating backing services as attached resources is that it's mostly taken care of by a proper config.
The twelve-factor app stores config in environment variables. See how to take wp-config.php and store configuration as environment variables.
The dependency system for PHP is Composer. In the world of Composer, the dependency declaration manifest is a composer.json file in the project root.
Your WordPress site/app codebase needs to be tracked in a version control system. Git is usually the de facto choice and you can't go wrong with it.