The procedure you describe does not produce a $T_2$-space in general. You can read [here][1] and [here][2] on MO, and also [this nice recent bachelor thesis by Bart van Munster][3]. 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][4].

**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 $\mathbf{Top}$.

  [1]: http://mathoverflow.net/questions/11191/nonhausdorff-dimension
  [2]: http://mathoverflow.net/questions/78175/largest-hausdorff-quotient
  [3]: http://www.math.leidenuniv.nl/scripties/BachVanMunster.pdf
  [4]: http://thales.doa.fmph.uniba.sk/sleziak/papers/