Tom, I think this is mistaken. Connes NCG setup does not rely on a manifold of any kind. In a spectral triple (H, A, D) the A is a * algebra of operators on a hilbertspace H and the D operator satisfies certain axioms. Nowhere is there a manifold.Connes still relies on a differentiable (spin-) manifold structure, otherwise he would not be able to introduce a Dirac operator; in that sense his approach does not solve any of the above mentioned problems, as far as I can see
It may seem as if Torsten thinks that a manifold is required, but his statement seems to me to be unclear. I would urge not to take his post as confirmation of the mistaken idea that a manifold is required in the definition of NCG.
Manifold IS involved in the definition of the special "almost commutative" case, where functions on a manifold are used in the construction. But we don't know that is is the necessarily the right way to go. There can be other ways to apply NCG. I doubt that "almost commutative" setup is the final word.