I'm seeing no L class availability on some AKL-SCL QF321 flights (eg 25 July). This could be why you are being directed via SYD to hook up with the QF direct flight. Perhaps other dates may give more favourable results?

An agent is likely to add a "booking fee" to the price - after all, their time costs money too. But, if you do the prep work, you might be able to negotiate a cheaper fee (ie tell them exactly what flights you want and know whether there is availability). Expert Flyer (subscription) will help tremendously.

Its segments and not cities. Check the rules, its 4 segments and IIRC its 6 in North America for the xONEx fares. Stopovers are a separate issue. Some cities have 2 or more airports (eg London). Moving between the airports counts as a segment. Stopovers are >24 hrs and transits <24 hrs - even 23 hrs 59 minutes. Transits can be used to your advantage.