Note that the embedding into the category of finite product-preserving presheaves only preserves finite coproducts, not arbitrary colimits.

By "cartesian category", I mean a category with finite products (I've clarified this in my answer). Yes, you take the limit in the category of small categories with finite products (which is equivalently the limit in the category of small categories).

I've updated my answer. Hopefully this clarifies the situation.

Downvoting because you didn't even include a title, which makes the paper impossible to identify now that the link is broken.

There is now an nLab page on this topic, with various references to papers on the subject.

@ArshakAivazian: yes, I should have mentioned that this doesn't just work for Set (Frei works with monads on an arbitrary base category, for instance). Glad I could help :)