For which topological measure spaces do open sets always have positive measure?

I am not sure if I understand your first question, but any topological space $X$ can be equipped with the Borel $\sigma$-algebra $\mathcal{B}_X$, i.e., the $\sigma$-algebra generated by the open sets of $X$. Then $(X, \mathcal{B}_X)$ is a measure space, which can be equipped with a Borel measure, which is a positive measure. However, there are Borel measures which are not "strictly positive".

The following might be an answer to your second question regarding the Haar measure:

The Haar measure is a translation-invariant Radon measure, which is a positive measure. Any non-trivial open set has, by definition, a positive measure with respect to the Haar measure. As you point out, the group structure is needed for the translation-invariance. The Hausdorff property is needed in order to prove the existence of the Haar measure. The book Locally Compact Groups by Markus Stroppel discusses properties of locally compact groups in the most general setting, and in general it is not assumed that the topological group is Hausdorff. However, in the proof of the existence of the Haar measure you will see that the Hausdorff property is in fact needed. In case you want to know where the Hausdorff property is exactly needed in the construction of the Haar measure, I highly recommend to read $\S12$ of Stroppel's book.