Originally Posted by JJeffrey

Trip #1: At LAX, you can use the LAX Flagship Lounge in T4 or the oneworld Business lounge in TBIT. The FL is generally regarded as the better one, but if you're arriving 5 hours before you have time to check both out.

There is no arrivals lounge at HND.

Departing HND you can use the JL Sakura Lounge or the CX lounge, both are very good.

Trip #2: Departing LAX, exact same as trip 1 above. At LHR departing on AA you will be in T3, you can use any of the oneworld business lounges there (AA, BA, CX, QF). CX and QF are hands down better than the AA/BA lounges.

Arriving at LHR, there is an AA arrivals lounge but you must be in business or first class or have EXP or PLT Pro status.

Trip #3: Departing LAX, exact same as trip 1 above. Connecting in LHR to FRA, once you get to T5 you can use any of the BA Galleries Club (business) lounges there. Same for the return, LHR-LAX on BA will depart from T5 (not T3) so same BA J lounges.

Trip #4: Exact same as trip 1 above, doesn't matter whether flying JL or AA.

Also you say you're flying with family, keep in mind oneworld rules only allow one guest in all situations above.