A hot dog, when prepared in the standard fashion, is not a sandwich. A sandwich necessitates two slices of bread* or other object. A hot dog bun is one continuous piece of bread with a slit where you insert the hot dog.
As for The Sandwich Alignment Chart, I'm a Structure Purist and fall somewhere between the Ingredient Neutral and Ingredient Rebel categories, though I lean more toward the Ingredient Rebel. Reason being? The first thing I think of when I think Ingredient Rebel is an ice cream sandwich.
* Now, ordinarily you associate a sandwich with ingredients of your choice placed between two pieces of bread. It's what my brain defaults to when I hear sandwich, something like PB&J or some such. However, I feel like something else can be used in place of the bread, like in the previous example of an ice cream sandwich being the cookies. Likewise, I am readily willing to substitute the traditional meat and veggies for whatever else. As long as it's something placed between two other things in the sandwich structural formation, it would be a sandwich.
In conclusion, if you so chose to place a hot dog on top of a slice of bread and then put another slice of bread on top of it, that would then be a sandwich. The thing that disqualifies a traditional hot dog from sandwichhood is the hot dog bun.