DURING A FLT AT NIGHT FROM MAJURO, MARSHALL ISLANDS TO CAIRNS, AUSTRALIA, THE PLT REQUESTED A DSCNT FROM FL 190 TO 12,000 TO REMOVE ICE FROM THE AIRFRAME. LATER, HE REQUESTED A CLIMB TO FL 200. AFTER STARTING THE CLIMB, HE MADE A MAYDAY CALL TO SIDNEY FLT SERVICE & RPRTD THE ACFT WAS IN AN UNCTLD DSCNT. NO FURTHER TRANSMISSION WAS RCVD FROM THE ACFT. THE ACFT WAS PRESUMED TO HAVE CRASHED AT SEA, ABOUT 370 MI ENE OF CAIRNS. NEITHER THE ACFT NOR THE PLT WAS FND. THUNDERSTORMS WERE RPRTD IN THE AREA & THE PLT RPRTD BEING IN THE CLOUDS DURING AN EARLIER DSCNT TO 12,000'. AN INVESTIGATION REVEALED THE ACFT WAS BEING FERRIED FM TULSA, OK. A PREVIOUS PLT, WHO FLEW IT TO OAKLAND, REFUSED TO CONT THE FLT DUE TO A CABIN PRESSURIZATION PROBLEM. A 2ND PLT, WHO WAS GOING TO CONT THE FLT, RTRND TO OAKLAND & LNDD THE ACFT IN AN OVER WT COND, WHICH RESULTED IN STRUCTURAL DMG. A 3RD PLT CONTD THE FLT, MAKING STOPS IN HAWAII & THE MARSHALL ISLANDS.