The way i see it, from my angle, (i can't tell because i'm not next to you) this should be resolved by first selecting the unwanted edge (edge mode) and press "backspace". I you select nothing or you select your whole mesh, well yeah everything will be deleted. If you can't select the unwanted edge, delete the surrounding…
It's normal, they are often left over vertices. If you want to get rid of them select the unwanted edges or vertices and press "backspace" key on the keyboard, as this will delete your selection but will leave everything around it untouched, contrary to delete that will remove shared geo. You can also look for floating…
I'm trying to do the backspace and it deletes my entire mesh. I can't even select the unwanted edges, it seems like they're connected, like if I select the smoothed vertex, it'll move the unsmoothed vertex as well. It's almost like it's showing the low poly cage when I smooth. Think '2' mode in Maya. Also, I'm still…
I suspect it's 2 point polys. Under that statistics tab, drop down Polygons, By Vertex, then hit the little + beside 2. That'll select all polygons with 2 vertices. You should be able to delete them as per normal.