If you want to maximize F flying HNL is a bad idea. I would take CX from the US to Hong Kong - 15 hours - long enough to eat, sleep, eat. CX F is superior to JL F. You could take JL to Europe at the end.
Same for LHR-NYC. The flight is too short. Go to LAX or SFO (BA, don't take AA) and then backtrack thru the US to NYC, then take CX to Asia.
AA has some flights LAX-MIA on the 777, i believe you would end up in one of the F seats.
You could swap HNL for some beach location in Asia (Thailand, Vietnam, Bali).
You could add two more sectors in Europe.

I am not sure what the cost of starting the ticket in TLV is, but check CAI, KRT and possibly even JNB (makes it more difficult to position, but you could sqeeze a second trip out of the ticket.