Re: Legal status of major supergroups
Well, in the real world, there's no such thing as "international police powers", as far as I am aware.
Aside from that, if my understanding of the situation in the US is correct, what you have described seems to be a bit of a mixture of FBI and CIA agent. I'm not particularly familiar with how the CIA interacts with the US armed forces, but presumably there are protocols of some description.
Depending on the number of supers there are around the place, it might be worthwhile separating the "police" and "intelligence" functions.
For Australia, I would probably attach my supers to the Australian Federal Police, and negotiate some kind of arrangement with the State and Territory governments to deal with the jurisdictional problems. Any overseas role would need to be negotiated with the relevant governments.