Questions 1, 3, and 4 have been very well explained in the other answers, but I have something to remark about Question 2.

Very frequently, objects that are meant to be like spaces will have some kind of algebraic data attached to them. But this algebraic data is attached contravariantly, that is, there's some functorial relationship between your category of objects and the opposite of the category of algebraic structures.

For example:

• Sets and Boolean Algebras. The power-set functor mentioned in Sammy Black's answer actually gives a contravariant functor from sets to Boolean algebras. This functor actually embeds the category of sets into the opposite category of Boolean algebras, so sets may be regarded as Boolean algebras with certain properties, except the maps go the wrong way.

• Schemes and Rings. A scheme is locally isomorphic to an object in the opposite category of commutative rings. In fact, the category of schemes admits a fully-faithful embedding into $Set^{Rng}$, the free cocompletion of $Rng^{op}$. This is called the "functor of points" approach to schemes.

• Compact Hausdorff Spaces and Unital C-Algebras. There's a contravariant equivalence between the category of compact Hausdorff spaces and the category of C-algebras with unit.

• Locales and Frames. A frame is a kind of distributive lattice, and is described in a completely algebraic way. It's space-like counterpart, called a locale, is studied in so-called "Pointless Topology" (don't laugh), and the category of locales is defined to be the opposite category of frames. This was inspired by the last example, which is:

• Topological Spaces and their Lattices of Open Sets. To every topological space, there is associated a certain lattice (the lattice of open sets). The requirement is that this association be contravariantly functorial - that is, every map of topological spaces must give rise to a map of lattices in the opposite direction. And that's what we have: a continuous map is one that induces a well-defined inverse-image map taking open sets to open sets.

So the idea that open maps seem to be more straightforward (so to speak) than continuous maps may be a common one, but in fact it seems that we get better categories of spaces if we ask the algebraic data to be contravariant.