Why are inverse images of functions more central to mathematics than the image?

I have a sequence of related questions:

Why the fixation on continuous maps as opposed to open maps? (Is there an epsilon-delta definition of open maps in metric spaces?)

Is there an inverse-image characterization of homomorphisms in algebraic categories? (What kind of map do you get if you look at a map from a group to another group, where inverse images of subgroups are subgroups?)

Inverse images have better set-theoretic properties than the image (for instance, commuting with unions, intersections, etc..) This clearly is a direct consequence of definition of a function. There is an asymmetry in the definition of a function (the domain and codomain behave differently with respect to the function). I think this also has consequences for differences between existence and uniqueness of left and right inverses for one-to-one or onto functions. Why this asymmetry? What are the historical reasons for the asymmetry? Whats sort of mathematics do we have if the definition of a function was purely symmetric? (For instance, f(a) may give multiple values, just like f^-1(a) may have multiple values).

Is it accurate to say the definitions for monomorphisms and epimorphisms in category theory correct for the asymmetry? (And hence, the notion of epimorphisms and onto-morphisms in concrete categories don't coincide)