Is Nominatim supposed to show "Point, ~0 miles" for all child streets?

The "Nominatim detailed view" is more or less a dump of the internal data Nominatim uses, so much of it is probably not very understandable unless you know how Nominatim works internally.

That said, to address your points:

• The "miles" indication is the distance to the object, not the object's size. So "~0 miles" in your example just means that this object is 0 miles from the main object of the page you are looking at. As the main object on the page you give is a polygon, it makes sense for streets that lie inside that polygon to have a distance of 0. If you look for example at the details for Düsseldorf, Germany, you see that most entries have a "miles" indication. That's because the main object is a point, and hence you can calculate the distance to other objects: http://open.mapquestapi.com/nominatim/v1/details.php?place_id=623958
• As to "Way" vs. "Point": I believe Nominatim internally does not distinguish ways and points, it just knows polygons and points. But that's a question only the Nominatim devs can answer definitively.