Now that my development is complete, my application can call Stack Exchange APIs successfully. Should I leave the StackApp post as it is and the app registration as it is? I'm looking to see if I need to do anything next on the OAuth app.
You should update your placeholder post to reflect the latest developments. You can use the template to know which information could be useful for potential users of your application. Otherwise, your placeholder question may be closed for the following reason:
This is a placeholder question for an app that's still under development; it will be deleted within 9 days unless regularly edited to show progress on development. If the app is ready for others to use, fill in the template and flag for moderator attention to indicate that this post can be reopened.
(The 9 days requirement isn't strictly enforced anymore, but the idea still holds.)