Can infinite first-order categories be specified other than as categories of models? I am glad to see that a general question like Is there a relationship between model theory and category theory? receives quite a lot attention and no down-votes for being too general and unspecific. So I feel encouraged to pose some follow-up questions (firstly restricted to first-order model theory).
Preliminaries
I hope the following statements are sufficiently sensible, precise and correct.

*

*Each first-order theory $T$ with signature $\sigma$ unambigously defines a class of (ZF-)models.


*This class of models of $T$ together with the $\sigma$-homomorphisms form a category (the category of models of $T$).


*Two first-order theories with two arbitrary signatures may define equivalent categories of models.

Definition: A first-order theory $T$
provides a model of a category $C$ if the
category of models of $T$ is
equivalent to $C$.


*

*Each category $C$ defines a (possibly empty) set of first-order theories: the set of all $T$ which provide a model of $C$.

Questions
[Remark: I had to work this question over, since it seemed to be ill-posed.]
Old version:
Can infinite concrete categories be specified
other than as categories of (ZF-)models of some (possibly higher-order) theory? Examples?
New version (explicitly restricted to first-order theories):

Given an infinite category of models
of a first-order theory $T$. Can this
category - or one equivalent to it - be
specified/represented/given
independently of any first-order
theory $T$ and its (ZF-)models?

Remark: $T$ of course can be specified/represented/given independently of its models: as a set of formulas.


Why is the notion of models of a (concrete)
category so uncommon? (Maybe because the
answer to the first question is "No"?)



Is there a genuine model-theoretic notion of two
equivalent theories if these have
two arbitrary signatures?


 A: As for the OP's last question, "Is there a genuine model-theoretic notion of two equivalent theories if these have two arbitrary signatures:"
Sure, it's called bi-interpretability.  See pp. 1378-1379 of this article for a definition of what this means for structures.  For theories, we say that T_2 is interpretable in T_1 if there is a family of interpretation formulas (as in the linked definition) such that for any model M of T_1, these formulas define an interpretation in M of some model of T_2.  Similarly, as above, we can define what it means for two theories to be bi-interpretable.
If you make the class of models of T into a category by declaring the morphisms to be the elementary embeddings (which seems very natural to me), then it follows directly from the definition that any two theories that are bi-interpretable (without parameters) have equivalent categories of models (via the natural "interpretation functors" which translate back and forth between the two languages).
.
A: Sure, by direct construction.  Rings, preorders, the category of paths of a given graph, etc.  But that's not what you wanted to know, is it?
A: Yes.  The category of topological spaces over the category of sets is not algebraic and therefore not a category of models over the category of sets.  It is a category fibered over the category of sets. The adjunction beteween the forgetful functor and its  left adjoint is not monadic because the left adjoint produces discrete spaces.  That is, the category of models over the monad induced by the adjunction is not equivalent to Top.  
Tom Leinster's book Higher Operads, Higher Categories has a description of this phenomenon on page 8.  He also gives a reference to Mac Lane's categories for the working mathematician, but you can find that yourself.
Edit: To answer your third second question, the reason that you don't hear people talk about models in the way you're asking is because they're called algebras over a monad or more generally algebras over an operad.
