A B767 CAPT RPTED NUMEROUS PROBS ON HIS FLT FROM DFW TO HNL; WITH AN INFLT EMER DIVERSION TO LAS.

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APPROX 30 MINS AFTER TKOF; THE COCKPIT WAS ADVISED OF AN ILL PAX; FEMALE AGE 72. A NURSE RELAYED THE PATIENT'S CONDITION AND STATS; WHICH WERE IN TURN FORWARDED TO THE ACR'S MEDICAL TEAM VIA DISPATCH. AS THE FLT PROGRESSED; HER CONDITION DETERIORATED TO THE POINT WHERE OUR MEDICAL TEAM ADVISED US TO DIVERT TO LAS. I USED CAPT'S EMER AUTH TO MAKE AN INFLT DIVERSION; DUE TO PAX ILLNESS. I ALSO USED CAPT'S EMER AUTH TO LAND OVERWT (APPROX 340000 LBS) AT LAS. IN BOTH INSTANCES; I DECLARED AN EMER WITH ATC. AFTER THE REQUIRED OVERWT AND ETOPS INSPECTIONS WERE COMPLETED IN LAS; THE FLT CONTINUED TO HNL. FATIGUE BECAME A FACTOR FROM THIS POINT ON. AS WE APCHED HNL AND BEGAN TO LOWER FLAPS; AN EICAS MSG; 'TE FLAP ASYMMETRY;' WAS OBSERVED WHEN FLAPS 20 WERE SELECTED. WHILE THE PROC FOR THE PROB WAS INITIATED; THE HNL TWR BADGERED US ABOUT LASHO OPS; EVEN THOUGH WE ADVISED THEM OF OUR PROB. I SHOULD HAVE DECLARED ANOTHER EMER; BUT FELT NOT ENOUGH TIME REMAINED BEFORE LNDG TO ANSWER QUESTIONS FROM THE TWR AND COMPLETE THE PROC. A NORMAL LNDG WAS MADE WITH FLAPS 20.