That's an interesting theory, but I don't think it's correct...

The reason is this... when an intersection happens on an edge, then it is only influenced by the two vertices that make up that edge - in your illustration, the neither the far left or far right vertices would have any influence (they'd have a zero weight value, either case).  So for morphing at least, averaging the weights of those two vertices would do no good at all (and I don't believe it's contributing to the lumpiness either - sorry).

I'll have to think more about your UV issue, but one thing to keep in mind is that UVs are for each vertex of every polygon and not just one UV per vertex (each vertex may have as many corresponding UV values as there are polygons connected to it).

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