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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.