I have fields in an object that have multiple record rules. When a user submits a form via pages on the front end of the database the record rules are working correctly. If we want to edit the information on that form on the back end via inline editing or on the front end, the record rules are no longer working to update the fields’ values. Do all of the record rules need to be duplicated again on the Edit form page as well? Or is there another step that I am missing?
I’m a bit confused with your terminology 
Record rules can only be applied to a form in the pages section for the live app, not a field in an object.
At the field level you can use validation and conditional rules but not record rules.
Record rules can also be set for live app in line editing but not for the builder environment.
If you are adding a record and have rules that run when the form is submitted, you may need to have similar rules on an edit form depending on what your app is doing.