Ahh Mangoboy - you ask lots of tough questions.
Righto - first things first. For the country that supposedly invented the Internet, the good ol' US of A has a few strange practices. First among them is their habit of breaking everything up into privately owned companies for the sake of efficiency and competition. Of course, these companies do not talk to each other as a matter of course, and this is why it is diabolically difficult to make a simple phone call over state lines.
One of their ideas is to run phone and power systems that are different to the rest of the world. There are lots of technical reasons why, mixed with history going back to the fight between Tesla and Edison. So the answer is no, your mobile will not work in the States. My advice - there is nothing you can do, come back soon to the land of OZ.
You should however be able to get Internet there. They still are one of the world's top users as a country, and have lots of the infrastructure that drives the net.
Listening to ABC Far North online is a little harder. Currently I am not aware of the local radio being streamed over the internet - so the only way I can think of is to
1) create a "skype" account or some other equivalent internet based VoIP provider
2) Ring ABC Far North (their numbers are on their
website3) Ask to be put on "hold" as they broadcast their show as their "music on hold"
Easy. Kind of. Well - not really, but it can be done

Cheers,
RR