As strange as it sounds, I'd still show the Qantas people in SYD your onward connecting flights. See if they can issue tags all the way. I don't think this is going to help so I'm going to suggest something else. Qantas probably does have an interline agreement with UAL, but I'm not 100% sure you can check your onward bags at the Bradly terminal (which is where I assume Qantas operates - although it may be in another terminal).

Go ahead and collect your bags and look for the hotel shuttles. Pick anyone you want - I'd pick the Marriott - go there and check your luggage with the bellhop. Hotels will generally store your luggage for a short time without charge (many hotels around LAX will balk at storing luggage for over a day). There really isn't much directly around any hotel near the airport, although there is a bus that can take you to the beach. I don't know much about it, but it exists. Also, there are inexpensive "flyaway" buses that take you to further points in/near LA. One goes to Van Nuyes. Not much to do there, but there are other buses and you can always rent a car there.

Or, I'd take advantage of the very low cost car rental companies (look at prices through Orbitz), in which case you could store your luggage in the trunk of your car and have transportation. I know switching driving sides can be difficult just after an international flight and that LA traffic can be daunting even for those of us who live there. But you can do it.

You won't really need an "international" driver's license as yours is in English.

Hope some of this helps.

Billy