Notice that my question is using the generic State Categories, not States.
For example, the results should include User Stories that have state Closed, but have any child work items that are not closed or removed. And it should include Features that are Closed, but have New User Stories. And more scenarios.
I want to be able to identify potentially incorrect status on parent work items.
We do not have a build-in filter State Categories in query editor. In other words, you have to use multiple states as a single state filter.
You can use In to search for any value in a delimited set. Separate values with the list separator that corresponds to the regional settings that are defined for your client computer. For example, you might use a comma(,).More details here.
Use filter options: Only return items that have matching links Return selected link types: Child to stand for parent and children relationship.
A sample query for your reference:
Moving to a closed state category will update the "Closed Date" field, otherwise it will be empty. You can setup a "Work item and direct links" query that has a parent and child filter that matches the criteria you expect.
