at.algebraic topology - Understand Cech Cohomology

For Question #2: Look at chapter 2 of Bott, R. & Lu, T. "Differential forms in Algebraic Topology", Springer GTM vol 82. There you will find a nice introduction to Cech cohomology through a Mayer-Vietoris general point of view.

For Question #4: Yes. Look at Bredon, G. "Topology and Geometry", Springer GTM vol 139, p. 289. There is a Mayer-Vietoris set-construction which is a nice way to avoid presheaf calculations.