Are concepts and properties studied in a category all preserved by morphisms? When study a category, are we only interested in those concepts and properties preserved by the morphisms, not those which cannot be preserved?
For example, in Terry Tao's blog

We say that one probability space ${(\Omega',{\mathcal B}', {\mathcal P}')} $ extends another ${(\Omega,{\mathcal B}, {\mathcal P})}$ if there is a surjective map ${\pi: \Omega' \rightarrow \Omega}$ which is measurable (i.e. ${\pi^{-1}(E) \in {\mathcal B}'}$ for every ${E \in {\mathcal B}}$) and probability preserving (i.e. ${{\bf P}'(\pi^{-1}(E)) = {\bf P}(E)}$ for every ${E \in {\mathcal B}}$).
...
In order to have the freedom to perform extensions every time we need to introduce a new source of randomness, we will try to adhere to the following important dogma: probability theory is only “allowed” to study concepts and perform operations which are preserved with respect to extension of the underlying sample space. (This is analogous to how differential geometry is only “allowed” to study concepts and perform operations that are preserved with respect to coordinate change, or how graph theory is only “allowed” to study concepts and perform operations that are preserved with respect to relabeling of the vertices, etc..)

A side question, do the probability spaces and the extension mappings form a category, so that my question at the beginning of this post can apply?
Thanks!
A somewhat related question is here, about the meaning of preserving mathematical structures.
 A: I first answer your latter question. This is clearly a category: the extendability is reflexive - just use the identity map, and transitivity you can easily show using the composition map.
W.r.t. your primary question, my knowledge of CT is very basic, e.g. I didn't have a chance to go beyond the first 4 chapters of Awodey, thus I am not sure whether I can give a complete answer. I'll try though. As Awodey mentions,

Let me begin with some remarks about category-theoretical definitions. By
  this I mean characterizations of properties of objects and arrows in a category
  in terms of other objects and arrows only, i.e. in the language of category
  theory. Such definitions may be said to be abstract, structural, operational,
  relational, or external (as opposed to internal). The idea is that objects
  and arrows are determined by the role they play in the category, by their
  relations to other objects and arrows, thus by their position in a structure,
  and not by what they "are" or "are made of" in some absolute sense. We'll
  see many more examples of this kind of thing later; for now we start with
  some very simple ones. Let me call them abstract characterizations. We'll
  see that one of the basic ways of giving such an abstract characterization is
  via a Universal Mapping Property or UMP.

As far as I understand this, it means that if you study categories, then you shall consider those statements that hold for all categories satisfying certain properties e.g. having finite products (which are defined only through morphisms and objects, not through the internal structure of categories). Alike in topology, the topological property of a topological space shall hold for all homeomorphic spaces. 
However, if you study a particular category -- say topological spaces, besides of applying purely categorial results which concern morphisms (e.g. UMP of products), you are also interested in certain topics which are relevant to this particular category. E.g. when the map is not only continuous, but is also open or closed.
Hope, it helps
