How can Jenkins help to implement CI/CD for Drupal?
What you've listed is actually two different things, continuous delivery and continuous integration.  When people talk about CI they typically are only referring to the testing part; automated unit, function, and integration testing with some sort of version bump or code promotion at the end.  Looks like the Jenkins wiki has some specific pointers about Drupal.

As far as environment promotion, you're getting into continuous delivery which Jenkins can also be used for.  Consider looking at the more modern Jenkins Pipelines for CD pipelines.