On the last question: no there should be no synonym from hausdorff-spaces to separation-axioms.
The issue is that somebody asking about a problem that happens to be placed in the context of a Hausdorff space might well use the former tag, while the question is not at all about separation axioms.
Now, one might argue the question then should not be tagged Hausdorff space to begin with, but that might be arguable even in theory and anyway in practice users just tag like that, and we should try to avoid synonyms that are likely to turn so-so tagging into tagging that makes no sense.
The same applies to separable-spaces. An existing question with that tag is a good example for the risk: Let $\mathcal{H}$ denote a separable Hilbert space. Construct a nonzero operator $T \in B(\mathcal{H})$ such that $T^2 = 0$.Let $\mathcal{H}$ denote a separable Hilbert space. Construct a nonzero operator $T \in B(\mathcal{H})$ such that $T^2 = 0$.
To tag this separable-spaces might not be needed or optimal tagging, but it is transparent why it was done and it makes some sense. Now, imagine it was tagged countability-axioms. It would make little sense to me.
I do not know if specific tags are needed. But if they are to be created, I'd recommend, at least in some cases, not not to name them like the objects. That is, one might use hausdorff-property or hausdorff-axiom rather than hausdorff-space. As explained above the point is to avoid them being used for questions where the objects merely happen to appear.
