Because of the alternating seasons, there is no clear 'low' season to Australia like Europe, for example. I think the best deals would be found in October/November and March/April but the price doesn't fluctuate much. The only time to avoid would be Dec 15-Jan 15 as this is peak travel time (Christmas, NYE + Australian summer holiday).
Without knowing your weather temperament I would recommend November as a good time to visit Sydney, it is springtime and generally not too hot, while still being comfortable and not too busy.
All airlines are fairly similar in cost and service. Are you flying business or economy?United and Qantas offer direct service to Sydney, or you could connect in LAX for Delta or Virgin Australia. You could also connect in AKL on Air New Zealand or in HNL on Hawaiian Airlines, both of which would reduce the longest flight time if that is a concern.
If business class is too expensive, Qantas offers a good premium economy product that would be a comfortable direct route option at a reasonable price.
Right now the USD is pretty strong compared to the AUD and makes a US visitor relatively comfortable, although Sydney is still not 'cheap'