For the first question. : The higher cohomology of a paracompact Hausdorff space (e.g. manifold) in the sheaf of continuous real valued functions is zero, because the sheaf is fine.
If I understand the spirit of your second question correctly, then you can use locally constant sheaves or more generally constructible sheaves to get interesting "topological" cohomologies.
A quick follow up: (1) The argument in the first paragraph also applies to sheaves of modules over the ring of continuous functions. (2) I don't know if every generalized cohomology theory (in the sense of algebraic topology) can or should be regarded as sheaf cohomology, but I leave to an expert to give a precise answer. On the flip side, I should point out that sheaf cohomology with arbitrary coefficients is generally not homotopy invariant, so it really is a different sort of beast.
