I'm flying first/business class from OMA to Cairns Nov. 12th and returning home on Dec. 03rd. All American flights. They have a 4 1/2 hour layover in LAX on the 12th. and a 3 1/2 hour layover in LAX returning on the 3rd. Would there be a cost ?

If they are international flights...because your flying 1st Class, you will have access to Admiral's Flagship Lounge gratis.

If either leg of an business/first itinerary is international you should have access in the connecting airport. This is also true for Exec Platinum on any ticket.

If the LAX to Cairns leg is business then you get free access in LAX. Im not sure if you would get it back from LAX to OMA as that leg is not intentional.