Theorem 1 is essentially the representation theorem (or the "representation definition" in Bourbaki) for the Alexandroff one point compactification of a T$_2$ locally compact space (the difference between continuous functions with value 0 at infinity and all continuous functions with any finite value at infinity does not change the dual, except for the delta measures concentrated on the point at infinity). And yes, the dual (for a compact T$_2$ space) is that of finite regular Borel measures or equivalently that of finite measures on the Baire $\sigma$-algebra, the minimum one that makes measurable all continuous functions (all such Baire measures are regular; Baire is smaller, being generated by the 0-sets of continuous functions i.e. closed=compact G$_\delta$ sets instead of all closed=compact sets).
It does not apply when there are not enough continuous (real / complex) valued functions, so you can apply it to a space whose associated T$_0$ is T$_2$ locally compact, but not in general. I think that Bourbaki has something in exercises for (possibly non T$_1$) quasi-compact and normal spaces.
To see easy examples of quasi-compact T$_0$ spaces with too few real or complex continuous functions, consider finite T$_0$ spaces, which are the same thing as finite posets (with principal order filters as basis for the topology).
Edit. Incorporating bathalf15320 absolutely correct comment:
if you want the dual of the space of continuous bounded functions on an arbitrary space, you note that by Gelfand duality you are considering the maximum, universal compactification of the T$_0$ complete regularization of the initial space, hence the dual is given by the Radon measures on that compact T$_2$ space. But the universal compactification is much more difficult to understand than the one-point compactification (the minimum one, which exists only for locally compact T$_2$ spaces).
As for T$_2$, already the old general topology book by Kelly noted (essentially) that what one needs in analysis is not T$_2$, but that the associated T$_0$ space (identify points that are in the same closed sets i.e. in the same open sets; example: in spaces of measurable functions, identify functions a.e. equal; more generally, for topologies and uniformities defined by a family of pseudo-metrics, one identifies points at 0 distance for all such pseudo-metrics) has a series of properties, in ascending order:
T$_2$: equivalent to unicity of limits;
T$_3$ (regular and T$_0$): equivalent to the previous one plus "the unique possible way to extend by continuity a function (i.e. extend taking the limit) always works giving a continuous function"
T$_{3+1/2}$ (T$_0$ and completely regular: points and closed sets are separated by a continuous real valued function, equivalently: topology induced by a family of pseudo-metrics, equivalently: by a uniformity): equivalent to embeddable as subspace of a compact T$_2$ space
(or also equivalent to "subspace of a normal T$_1$ space").
[Having more open sets than a T$_{3+1/2}$ topology is equivalent to "distinct points are separated by a continuous real valued function"; it is independent from T$_3$ and lies between T$_2$ and T$_3+1/2$,
but usually when one talks about "sufficiently many continuous functions" means the stronger condition T$_3+1/2$.]
Now, sometimes one defines "locally compact" as "each point has a basis of quasi-compact closed neighborhoods", which is the same as "regular and each point has a quasi-compact neighborhood" or also "the associated T$_0$ space is locally compact T$_2$". [Kelly's book used very much the "regular" trick to avoid Hausdorff].
