Here's a
oneworld option from my hometown BNE.
The criteria (i.e. check-in after 5pm Fri and back by 9am Mon) did make it hard to design a good
oneworld itinerary from Australia. There might be some flexibility if you started in SYD or MEL, but the long and short of it is that there are not a lot of ultra long haul flights that leave after 5pm from BNE, SYD or MEL. FWIW, all of the Trans-Pacific flights from East coast Australia leave before midday; AKL-LAX is a silent exception but getting to AKL from Australia to make the AKL-LAX flight at around 1450h or 1550h violates the "must check-in after 5pm rule".
Code:
Day Flight Sector Dep Arr Dur'n Miles A/C Connect Remarks
FRI QF 635 BNE D-MEL T1 1755 2020 2:25 855 738 2:30
FRI QF 29 MEL T2-LHR T4 2250 1330+1 21:55 10517 744 2:30 stop HKG 1:45
SAT BA 269 LHR T5-LAX TB 1600 1900 11:00 5449 744 4:20
SAT QF 16 LAX T4-BNE I 2320 0610+2 13:50 7177 744 END
Total time in flights is 49 hours and 10 minutes, covering 23,998 miles. 9 hours and 20 minutes is spent in transit.
Check-in at BNE needs to be 30 minutes in advance with checked baggage. Without checked baggage it is 15 minutes although I'm unsure if that still works if you have to check in for an international connection. (If I had to guess, I'd say it'd be alright.)
In any case, that means arriving at the airport at about 1725h. Unless you were a speed demon with perfect traffic conditions, you'd probably need an early leave pass from work to get to the airport in time for check-in.
There's enough breathing room at each stop (including the HKG stop) for a shower, a stretch, a quick bite to eat if on-board wasn't enough or good enough and check a few emails or hit Twitter, etc.

Well, except for perhaps LHR with the T4->T5 terminal transfer and only 150 minutes....
This is a fairly pedestrian itinerary, but given the criteria there was no room to squeeze in more flights, except for perhaps LHR-JFK-LAX or the like, but the search engine I was using couldn't find anything which made the LAX-BNE flight in time. It fits the rules of a
oneworld Explorer, but I had to construct this dummy itinerary with the availability engine turned off. Especially with the availability engine turned on, it took me forever (and sometimes could not) get the BA269 flight. Also, this itinerary should be achievable with MEL in DST; the arrival time in MEL should change to 2120h and the connection time to QF29 would be 1h 30m.
The going price for this itinerary would be AUD 11,340 in J (DONE4), and that's without taxes. You can't book this as a LONE4 because of the minimum 10 day rule (you could probably book each sector separately in Y and still end up cheaper than paying for a DONE4). In any case, not a great mileage run except for kicks, giggles and status (of a different kind, i.e. to feed your office ego

).