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].


  [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/