The procedure you describe does not produce a $T_2$-space in general. You can read here and here on MO, and also this nice recent bachelor thesis by Bart van Munster. But you do get the Hausdorffification by iterating that procedure (transfinitely many times, in general). It follows that $T_2$-spaces form a reflective (full, replete) subcategory of $\mathbf{Top}$. There is no co-Hausdorffification though, because the only subcategory of $\mathbf{Top}$ that is simultaneously reflective and coreflective is $\mathbf{Top}$ itself: see Kannan, Reflective cum coreflective subcategories in topology, Math. Ann. 195 (1972), 168-174, and also the 2006 PhD thesis by Martin Sleziak, available here.
Added: Apologies if I misread the question by assuming the factorization to be unique. In order to partially rehabilitate myself let me add some comments. The full subcategory of $T_i$-spaces happens to be reflective for $i=0,1,2,3$ (Kennison characterized reflective subcategories of $\mathbf{Top}$ as those full subcategories which are closed under products and subspaces in $\mathbf{Top}$, see his paper "Reflective functors in general topology and elsewhere", Trans. Amer. Math. Soc. (1965), 303-315). This means that for those values of $i$ there exists a notion of $T_i$-ification (and the factorization is unique). Regarding the weak co-$T_i$-ification (let us call the corresponding weak reflector $WR_i$), at least for $i=1,2,3$ it does not exist by the argument in Eric Wofsey's beautiful answer. If you prefer a published account you can consider Lemma 5.5 in Herrlich's paper "Almost reflective subcategories of $\mathbf{Top}$", Topol. Appl. 49 (1993), 251-264, which states:
Lemma 5.5: If $\alpha$ is an infinite cardinal with successor $\alpha+1$, then there is a continuous map $f\colon 2^{\alpha+1}\to S$ from the $\alpha+1$ power of the discrete 2-point space $2$ onto the Sierpinski space $S$, which does not factor through any $T_1$-space $X$ with $\mathrm{Card}(X)\leq\alpha$.
Then proceed as follows: if such a weak co-$T_i$-ification $WR_i$ existed, take $\alpha$ to be the cardinality of $WR_i(S)$. Observe that $WR_i(S)$ is $T_1$ and $2^{\alpha+1}$ is $T_i$, $i=1,2,3$, being a power of a $T_i$-space, and apply the lemma.
I would be interested to know whether $T_0$-spaces are weakly coreflective in $\mathbf{Top}$.
