Let me extend my comment above to give some more detailed information, especially regarding metrisability (I am not aware of any “theory: First of all it is well-known that a topological group is metrisable if and only if it is first-countable. However, such a metric might not have nice properties. However, a locally compact group is metrisable by a left-invariant, proper (every bounded closed set is compact) metric if and only if the group is second-countable, see the papers Metrics in locally compact groups and Proper metrics on locally compact groups, and proper affine isometric actions on Banach spaces. Properness of the metric implies completeness, thus every second-countable locally compact group is actually a Polish group.
Actions of Polish groups have been studied extensively in descriptive set theory, especially continuous actions on Polish (completely metrisable) spaces, but as usual in descriptive set theory also cases of measurable actions with respect to some $\sigma$-algebras. Gregory Hjorth has written a book Classification and Orbit Equivalence Relations and a shorter chapter A survey of current and recent work on the theory of Borel equivalence relations for the Handbook on Set Theory, which appeared recently (this topic has been applied to C*-algebras and dynamical system theory, too, I am no expert, but I am mentioning it since you mention these topics in your profile). Locally compact Polish groups are kinda the simplest case in this theory, local compactness guarantees that the group actions are not “turbulent”, their orbits are $\Sigma^0_2$-sets (countable unions of closed sets). I do not know what you do expect exactly, maybe this theory is useless for you, but it might be reasonable to notice its existence. Invariant Descriptive Set Theory by Su Gao is another recent monograph covering Polish group actions extensively. Let me quote a result from that theory that might give you some impresion of what is known about the general structure of such group actions:
Given an action of a locally compact group acting continuously on a Polish space $X$. Then there exists a Polish space $Y$, a Borel measurable equivalence relation $E\subset Y\times Y$ such that every equivalence class is countable and a Borel measurable function $\theta\colon X\to Y$ such that for all $a,b\in X$ we have $Ga=Gb\Leftrightarrow \theta(a)E \theta(b)$.
Using a web search I have also found this document considering left-invariant, proper metrics on locally compact groups which do not generate the topology. Under that condition you can also consider some non-second-countable groups (sorry, I did not notice the comment which already referenced this).