Shall we assume that "surjects" refers to a surjective homomorphism, or does it mean any surjective map onto ℤ ?

How can the "how many" questions make sense without any reference to size or distance? And by "inside" are you referring to the bounded portion of the complement of the sphere in euclidean space, or something else? (I also notice that the dimensions of the sphere and of the euclidean space are left unmentioned.)

Taking the union of all the line segments from some given point P on the curve to each of the other points on the curve almost gives such a surface. Perhaps that idea can be refined to always work? (Possibly by a judicious choice of P.)

Yeah, 6 seems to be the size of the shortest polygonal trefoil, but can it be arranged that the TSP solution is this trefoil?

Thanks for editing — it looks much better now.