Is it possible to unwrap genus-0 mesh so each vert has one and only unique uv? To unwrap a genus-0 mesh such as a sphere, I can make a full sphere from an unwrapped half sphere applied with mirror modifier. the amount of uvs equal the amount of vertices.  But the two verts selected in image below have two different uvs with the same value.(by checking in the exported .obj)

Or I can make a seam then 'Unwrap'. This way the number of uvs is larger than the amount of vertices. Each uv coord is unique, no repeated values.

I wonder how to unwrap a genus-0 mesh so each vert has one and only unique uv? Unique means there're no repeated uv coord values, it's not unique if the ith vt and jth vt have the same coord values.
No.

Unwrap. Meaning peeling the surface off of a solid and laying it out flat on the ground. 

If your requirement is that there are no duplicated texture coordinates at all, it would mean that you are trying to peel an orange without cutting its skin. Even if you manage to "beam off" the skin in one piece you end up with a hollow shell that you cannot lay out flat on the ground without the front and back sharing the same area. 

And the moment you cut it you get seams and therefore more uvs than vertices. if you merge them again later you once more end up in the situation described above.

This is true for all solids, not just genus-0 ones like a sphere.