Category theory and model theory as "natural" counterparts

I'm not an expert in model theory anyway I'll try to answer your questions.

From what I get your problem come from the fact that both model theory and category theory are related with the study of stuctured objects and morphisms between them.
There are categories which aren't at all build up from structured objects and morphisms stucture-preserving, for instance monoids, groups and posets are categories too, and seeing this objects as categories is useful for some applications.
Model theory instead deal exactly with models of a theory which are exactly stuctured objects and the stucture preserving morphisms, so it deals with categories of models of given theories (to be exact if I'm not mistaking, model theory also deal with theories' morphisms and derived morphisms between theories' models, but also this can be seen in terms objects and morphism).

After this not too short introduction let's try to answer your questions:

Answer #1: I suppose that the textbook you are referring to were written in time when the deep connection between model theory and category theory weren't well known. Try to take a look to book about categorical logic.

Answer #2: As I said above categories can be viewed as models of a particular (first order) theory, by the way this is not really useful because of the size issues I mentioned above. By the way category theory via notions of categories (with enough structure), functors (preserving the said structure) and natural transformations offer a new way to define the notion of theory, model and model transformation. In this way it become possible to study the notion of model of a theory in any category, where classical model theory become simply the study the theory of models in $mathbf{Set}$, the category of sets and functions.

Answer #3:I don't know if there's any satisfactory answer to this question, mostly because as I said category theory and model theory are really different theories which aims to study different objects (the first one deal with theories and models, the second with categories, functors and natural transformations, but also other objects if we consider higher category theory as category theory).
Maybe it could be more interesting studying the relation between classical (i.e. set theoretic) model theory and categorical model theory, but I don't know enough to talk about this.

Answer #4:If by level of abstraction you mean if one can be consider as a special case of the other I guess the answer is yes and no: you can build a first order theory of categories, functors natural transformation but from another point of view model theory can be completely rephrased in categorical term. Seeing from this point of view the question seems to me very similar to the chicken or the egg causality dilemma, and I don't think it's really useful this point of view, I would never consider group theory just as the study of the models of the theory of groups. :)

I hope this helps.