1. Does the data you get have a 'string' name along the triangle edge?
No, unfortunately not.
2. P-string would solve this.
I have done this, but the triangulation derived from the P-string is not as "pure" as the original, because the break line influence / restrictions are lost. On "natural" surfaces, it could be acceptable, but man-made surfaces such as existing roads are not modeled as accurately as they would have been, using 3-d strings.