I don't think AA.com shows CX awards if you're looking for that. I don't think Alaska's website does either so you have to call to book those.
I use awardnexus.com when I am searching for availability but I think you have to pay to use it (I am grandfathered in as a free user).
It's been my experience that you have to plan very far in advance (like as soon as the schedule is released) for awards to that part of the world. But things can change frequently.
You can transfer your UR points to United depending how many you have in the bank. I prefer to avoid flying UA metal so my recent trip took a very roundabout routing via NRT and BKK.
You might also look at transferring MR and/or UR points to Virgin Atlantic and use that program to book on Virgin Australia but their redemption costs are pretty high.