Example Wanted: When Does Cech Cohomology Fail to be the same as Derived Functor Cohomology?

Q1: A very simple example is given in Grothendieck's Tohoku paper "Sur quelques points d'algebre homologiquie", sec. 3.8. Edit: The space is the plane, and the sheaf is constructed by using a union of two irreducible curves intersecting at two points.

Q2: Cech cohomology and derived functor cohomology coincide on a Hausdorff paracompact space (the proof is given in Godement's "Topologie algébrique et théorie des faisceaux"). I don't know of an example on a non paracompact space where they differ.

In the paper Pathologies in cohomology of non-paracompact Hausdorff spaces, Stefan Schröer constructs a Hausdorff space which is not paracompact, and for which sheaf cohomology with values in the sheaf of germs of $S^1$-valued functions does not agree with the Čech cohomology (for example - the same is true for other sheaves).

The space is constructed by taking the countably infinite join of disks $D^2$, with the CW-structure consisting of two 0-cells, two 1-cells and a single 2-cell, using one of the 0-cells as a basepoint. Then he takes a coarser topology, whereby the open sets are open sets from the CW-topology, but only those that either don't contain the basepoint, or contain all but finitely many of the closed disks. With this topology the space is not paracompact, but is $\sigma$-compact, Lindelöf, metacompact.... and contractible! Sheaf cohomology is non-trivial however.

An interesting point to note is that this is not a k-space, and the k-ification of this space is the original CW-complex.