It is oversimplified, I know, but just as a superficial analogy, one may think of the fact that abelianization of the fundamental group is the first homology group, as some remote relative of class field theory. To attempt increasing resemblance, one may dualize and speak about correspondence between characters/one dimensional representations of the fundamental group over some field and first cohomology with coefficients in the multiplicative group of that field; in sufficiently nice cases this should in particular give correspondence between appropriate brand of coverings, line bundles and their Chern classes. One may even try to imitate ramification by considering ramified coverings and not necessarily locally free coherent sheaves or I don't know what (Question zero - has this actually been made more rigorous and if yes, where?).

I am not aware of any relevant higher-dimensional stuff, relating, say, 2-dimensional representations of the fundamental group to characteristic classes of the corresponding vector bundles in any context - say, over $\mathbb C$ even, let alone other coefficient fields or higher dimensional non-locally-free coherent sheaves. Are there any texts about this?

The question What is the precise relationship between Langlands and Tannakian formalism? seems to be related but I am after something much more simple-minded