Why are inverse images more important than images in mathematics?
Continuity is important not because of its inverse-image-ness, but because the definition corresponds to the geometric notion that it's intending to capture. A continuous function "takes close things to close things". This geometric intuition precedes the notion of open sets, and even the epsilon-delta definition of continuity.
I think the same sort of thing is true of functions in general. The asymmetry is there as a consequence of the idea that the definition is intending to capture; functions weren't originally thought of as a special class of relations, they were thought of as "rules" for manipulating numbers, and the idea of f(x) being unique and f${}^{-1}$(x) not being unique is the only natural way to capture the idea of a "rule" in a more general context.
I don't know that the notions of monomorphisms and epimorphisms really "correct" for the asymmetry, but I don't think it's something that ought to be corrected anyway. Monomorphisms of sets are the same as injective functions, and epimorphisms of sets are the same as surjective functions, and that's the way it should be. In categories like Ring where, say, epimorphisms aren't always surjective, it's because the non-surjective epimorphisms, in some sense, are "surjective as far as ring maps are concerned"; for example, the inclusion of $\mathbb{Z}$ into $\mathbb{Q}$ is epi because a map out of $\mathbb{Q}$ is determined by what it does to $\mathbb{Z}$. I don't think this has much to do with the asymmetry you're describing.
Open sets can be identified with maps from a space to the Sierpinski space, and maps out of a space pull back under morphisms. (In other words, if you believe that the essence of what it means to be a topological space has to do with functions out of the space, you are privileging inverse images over images. A related question was discussed here.) I think essentially this kind of reasoning underlies the basic appearances of inverse images in mathematics. For example, in the category of sets, subsets can be identified with maps from a set to the two-point set, and again these maps pull back under morphisms. This should be responsible for the nice properties of inverse image with respect to Boolean operations.
Your third question was asked, closed, and deleted once; I started a blog discussion about it here.
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.