That LineString (in A) will contain the actual coordinates of the
streets themselves (each corner and bend). (and the routing program
must of known about them to be able to export)
I dont think there is any KML option to make GE 'snap lines to roads',
it will always just plot the coordinates as provided in the file.
On placemarks contained in a KML file, you can open the placemarks in
GE, and click the Directions to/from here, so thay you end up creating
a routing between any two placemarks. (and you can then copy that to
your file)
Or you could write a program that read your file and then added
routing,
either i) grabbing data direct from Google Maps and incorperating it
in your file - but I dont know the legality of this,
or ii) or just providing network links that 'hotlinks' the data into
your file - this I think is ok, while its a gray area, its not
actually copying the data, just using it - just like you would doing
it manually