Cogenerator of Categories of Topological Spaces Satisfying Some Separation Axiom

Perhaps I should have been a little better at Googling before posting this question, but it seems to be answered, to a degree, in a paper from 1980 by Giuli, cited below. In particular, any epi-reflective subcategory, i.e. one that is closed under products and subspaces (hence whose inclusion is limit preserving), has a system of cogenerators that do not form a set. This property is referred to by Giuli as being "weakly initial." This, if I'm reading this right, looks like a system of spaces $X(\Lambda)$ for every cardinal $\Lambda$. Now, in some nice cases, e.g. $T_0$, $T_4$, Tychonoff, and compact Hausdorff, there's in fact a single cogenerator. But this is not the case (according to that paper, though not proven) for $T_1$, $T_2$, $T_3$ or Urysohn spaces. In fact, Giuli says that there are not even systems of cogenerators known for $T_2$, $T_3$, and Urysohn (or $T_{2\frac{1}{2}})$.

However, Giuli does confirm that the interval $[0,1]$ is a cogenerator for Tychonoff spaces and that the Sierpinski space is a cogenerator for $T_0$-spaces (apparently). All of this discussion begins around Theorem 1.2 of that paper.

Giuli, Eraldo, Bases of topological epi-reflections, Topology Appl. 11, 265-273 (1980). ZBL0441.18012.

Concerning your third question, the cogenerators of the category of general topological spaces are precisely the non-$T_0$-spaces. See Example 7.18 Remark (4) in Adamek, Herrlich and Strecker's Abstract and concrete categories: The joy of cats (pdf).

The answer to your question (0) is "no". The Wikipedia page is in the context of a category with zero object, in which case a zero morphism is one that factors through the zero object. Consider a category with four objects $A,B,C,0$, with $0$ the zero object, and four nonzero nonidentity morphisms $f,g:A\rightrightarrows B$, $h:B\to C$, and $k:A\to C$, where $h\circ f = h\circ g = k$. Then $C$ is a cogenerator in the Wikipedia sense, but $\hom(-,C)$ is not faithful.

The Wikipedia page seems to believe that the two definitions are equivalent at least in an abelian category, since the subsequent sections use instead the property (equivalent to the usual "faithful represented functor" definition) that every object injects into a product of copies of the cogenerator. I'd be surprised if this were true; I would only expect defining (co)generators to detect triviality of objects rather than morphisms to work in contexts like a triangulated category, where whether a morphism is an isomorphism can be detected by whether its cone is zero. The most I can prove from the Wikipedia definition in general is that $\hom(-,C)$ reflects epimorphisms.