Why don't we take clopen maps as morphisms of the category of topological spaces? Here, by clopen maps I mean a function mapping an open set into an open set and a closed set into a closed set. We say continuous maps are the morphisms of the category of the topological spaces. But isn't it more natural to consider clopen maps instead of continuous maps? For example, we say group homomorphisms are the morphisms, and a group homomorphism $h:G\rightarrow H$ preserves the group structure of "$G$" in "$H$". On the other hand, in some sense, a continuous map $f:X\rightarrow Y$ preserve the topological structure of "$Y$" in "$X$". The direction is reversed. I am wondering if there is any good explanation for this. Thanks!

Because then $f : \mathbb{R} \to \mathbb{R}$, $x \mapsto e^x$ would not be a morphism (its image is not closed). Recall that almost every notion in mathematics is motivated by examples, and this includes the definition of a category and explicit examples of categories. You don't want to just play around with the axioms, you want to study the examples you are actually interested in. And a definition of a morphism of spaces which does not include the exponential function is certainly not natural.
Another approach to topological spaces is provided by Kuratowski spaces. Here, one has a relation $x \prec A$ between points $x$ and subsets $A$ of the underlying set, which is supposed to mean that $x$ lies in the closure of $A$. This relation $\prec$ has to satisfy some axioms. Then, a map $f$ is continuous if and only if it preserves this relation $\prec$:
$$x \prec A \Rightarrow f(x) \prec f(A)$$
You might also want to have a look at frames; here the morphisms are really algebraic homomorphisms as you suggest. Frames resp. locales are a generalization of topological spaces. This is also called "pointless topology".
