It is a tricky notion. Given a homotopy type $X$ one can construct its complex schematization $(X\otimes \mathbb{C})^{sch}$. By definition this is a higher stack on the etale site of schemes, which has the property that its fundamental group is the pro-algebraic completion of $\pi_{1}(X)$ and every finite dimensional representation $V$ of $\pi_{1}(X)$ gives rise to a coherent sheaf on the schematization, so that the cohomology of the local system $V$ on $X$ is naturally isomorphic to the cohomology of the coherent sheaf on the schematization. The schematization exists by a theorem of Toen.

The holonomy does not provide such a bijection. It only provides a bijection between the space of flat connections on the trivial bundle and a connected component of of the representation variety. Often the representation variety will have more components. If $M$ is a compact manifold and $G$ is a complex reductive group, then this bijection is a homeomorphism. In fact if $M$ is a complex projective algebraic variety, the bijection is a complex analytic isomorphism.

It doesn't have to be locally free. If $p$ is a point on a smooth curve $C$, then $\mathcal{O}_{C}(-p)\oplus \mathcal{O}_{p}$ has trivial first Chern class.

The fact that the minimal resolution of the compactified relative Picard is again a K3 can be checked directly from Kodaira's classification of singular fibers. The type of the singular fiber is determined by the local monodromy and the local monodromy doesn't change under dualization because it is in $SL_2$. Once you know that singular fibers are the same as the original ones, the canonical class formula for an elliptic surface tells you that you have a K3. This doesn't work in higher dimensions already topologically as Mark explained above.