ct.category theory - What is the opposite category of the category of modules (or Hopf algebra representations)?

I am three months late for the party, but I'll still add an answer (to the first question):

There is the notion of "locally finitely presentable category" which can be described equivalently as

(i) a category such that there exists a set of finitely presentable objects such that every object is a directed colimit of these.

(Explanation: $X$ being finitely presentable means that $Hom(X,-)$ commutes with directed colimits, for modules it is equivalent to the usual notion of finitely presented, intuitively because for a given morphism into the colimit each of the finitely many generators has to go into some finite stage of the colimit diagram, then one can form a cone over these finitely many objects, as the diagram is directed, and the morphism factors through there)

(ii) a category of all models in SET of a finite limit sketch

(Explanation: A finite limit sketch is a small diagram, or small category, with distinguished cones, a model in SET is a functor from this small category to sets which maps the distinguished cones to limit cones - example: the limit sketch of groups is a category $C$ with an object $G$, a map $G times G rightarrow G$ a commutative diagram stating associativity etc and the cones which e.g. say that the object $G times G$ is the product of $G$ with itself. A group is a functor from this to SET which takes $G times G$ to the product, i.e. maps the distinguished cones to limit cones. Likewise for a ring $R$ the limit sketch for $R$-modules is given by an abelian group object $G$ as above plus one group endomorphism of $G$ for each element of $R$, behaving as dictated by the multiplication in $R$)

By either description you see that the category of $R$-modules is a locally finitely presentable category and now there is a theorem (e.g. Adamek/Rosicky Thm 1.64) saying that the opposite of a locally finitely presentable category is never itself locally presentable unless we are in a trivial case where our category is a poset.

The cool thing about this result is that it not only answers your question about modules, but also applies to all other algebraic structures (the theory of locally presentable cats gives a characterization of categories of algebraic structures) and I like to read it as is a provable mathematical statement which reflects the duality between algebra and geometry: The dual of an algebraic category is not itself algebraic, but geometric...