Sadly no, we are too far from the US to get direct flights (It would be 18-19 hours or so). They all connect in Brisbane or Sydney.
You will see a lot of Asian flights to and from Australia.
Darwin is 24 hours - so any time is good. If you are more interested in tracking airlines instead of just GA aircraft, for you between 5:30am and 12:30pm will be good. However, our busy period is between 3:30pm and 3:30am your time. Airlines will only make up a small part of the traffic (because there are heaps of light aircraft).