Is a hot dog a sandwich?
No. It’s in the subcategory of bread based meals like sandwiches and pizza, scones, etc. The topology is different. Sandwiches are built with layers. Hot dogs have a bread cup/container with extra condiments wedged on top.
Hot dogs also require a sausage. Sandwiches are agnostic to sausages.
If the bun splits along the fold into two separate halves, does the hotdog become a sandwich?
Well, if you’re riding your bike and a wheel falls off does it become a unicycle?
I’m not sure that’s a valid analogy. Losing an entire wheel would be more akin to losing one half of the bun.
The bun has lost nothing so material. The closest bike analogy might be a snapped brake or gear line, but even that might be too much, because the break in the hotdog does not significantly increase its danger level.
