TITLE: Category of categories as a foundation of mathematics
QUESTION [30 upvotes]: In

Lawvere, F. W., 1966, “The Category of
  Categories as a Foundation for
  Mathematics”, Proceedings of the
  Conference on Categorical Algebra, La
  Jolla, New York: Springer-Verlag,
  1–21.

Lawvere proposed an elementary theory of the category of categories which can serve as a foundation for mathematics.
So far I have heard from several sources that there are some flaws with this theory so that it does not completely work as proposed.
So my question is whether there is currently any (accepted) elementary theory of the category of categories that is rich enough so that one can formulate, say, the following things in the theory:

The category of sets.
Basic notions of category theory (functor categories, adjoints, Kan extensions, etc.).
Other important categories (like the category of rings or the category of schemes).

The elementary theory I am looking for should allow me to identify what should be called a category of commutative rings (at best I would like to see this category defined by a universal 2-categorical property) or how to work with this category. I am not interested in defining groups, rings, etc. as special categories as this seems to be better done in an elementary theory of sets.
P.S.: The same question has an analogue one level higher. Assume that we have constructed an object in the category of categories (=: CAT) which can serve as a, say, category C of spaces. Classically, we can associate to each space X in C the sheaf topos over it. In the picture I have in mind, one should ask whether there is a similar elementary theory of the category of 2-categories (=: 2-CAT). Then one should be able to lift the object C from CAT to 2-CAT (as one is able to form the discrete category from a set), define an object T in 2-CAT that serves as the 2-category of toposes, and a functor C -> T in 2-CAT.

REPLY [22 votes]: I've just run into your question now. I realized it's over a year since it was asked, but since the question is not closed and I happen to have at hand some references (as I'd been interested in such developments for a while), perhaps you should take a look at the following proposals.
Lawvere's original paper was indeed flawed, as reviewed by Prof. Isbell. Something later known as "category description theorem", a way to generating objects in the theory (i.e. categories) from some description of its intuitive structure, happened to be non provable in the original theory. A thorough review of the paper including ways to fix this and save a considerable portion can be found in: Blanc G., Preller A., Lawvere’s basic theory of the category of categories. Journal of Symbolic Logic 40 (1975, no. 1), 14–18 (doi:10.2307/2272263, JSTOR).
Subsequent proposals also try to address the point of the "category description theorem" by taking a variant of it as an axiom. This is done in Blanc G., Donnadieu M. R., Axiomatisation de la catégorie des catégories. Cahiers de topologie et géométrie différentielle 17 (1976, no. 2), 1–35 (Numdam).
Finally, McLarty proposed a clean, elegant and well presented axiomatization where proofs of independence and relative consistency are given. This can be found in McLarty, C., Axiomatizing a category of categories. Journal of Symbolic Logic 56 (1991, no. 4), 1243–1260 (doi:10.2307/2275472, JSTOR). McLarty theory was meant to be taken in conjunction with certain axioms for specific categories or functors needed for specific purposes.
But I believe that all these three proposals are able to formulate the three bullets you mentioned in the first part of your question. For some other purposes one should take a closer look at them.