So there is a hell of a lot of info on mileage programs

But what is the consensus on collecting miles and tier status? for a DONEx ticket like below?

CAI-xLHR-oARN-oLHR-oMEX-oMIA-oJFK-oLAX-oHNL-oNRT-oHKG-oMLE-oHKG-oBOM-oDOH-oDXB

Should I go for the AA challenge? and/or can I cherry pick and selectively give my AA, BA or CX membership number depending on which carrier&sector I am flying?

Thanks for helping!!!!