I guess some of my problems come from the fact that I can hardly visualise cohomology. For homology in dimensions 1–3, I claim to have at least some intuition what homology classes look like, and when two cocycles are homologous. But for cohomology, I mostly have no idea how the cohomology classes, once I look e.g. at a CW- or $\Delta$-complex, look like.
For $H^1$ of manifolds with a $\Delta$-complex structure, I know that I can think of cocycles as closed loops crossing transversally through edges, assigning the value 1 to all edges they cross, and 0 to the others. How does this idea extend to non-manifold complexes, e.g., if more then two 2-cells are adjacent to a 1-cell? How do cocycles look like if they represent non-trivial cohomology? How do cohomologous cocycles look like? And how about $H^2$…?