Since I am travelling in coach so I don't have much consideration about 2-class/3-class flight, you may want to consider the following with your comfort.
NA:
LAX-SJU (longest transcon, but 2 class only)
DFW-ANC (longest segment, but 2 class only)
to fullfil your LAX/ORD req.
something like HKG-JFK-DFW-ANC-DFW-ORD-SJU-
LAX-LHR
Europe: -LHR-IST-LHR-DXB-LHR-
A tools you can use is download the AA time table and type in the big city pairs, sometime it will have 2 segments in one flight number, i.e. counted as 1 segment in RTW rule. (e.g. LAX-SJU)